Category Archives: Arithmetic

Lime Time

by in Arithmetic, Circles, Fractals, Geometry, Integer Sequences, Mathematics and tagged , , , , , , , , , , | Leave a comment

What do you get if you list every successive pair of entries in this sequence?

1, 2, 1, 3, 2, 3, 1, 4, 3, 4, 1, 5, 2, 5, 3, 5, 4, 5, 1, 6, 5, 6, 1, 7, 2, 7, 3, 7, 4, 7, 5, 7, 6, 7, 1, 8, 3, 8, 5, 8, 7, 8, 1, 9, 2, 9, 4, 9, 5, 9, 7, 9, 8, 9, 1, 10, 3, 10, 7, 10, 9, 10, 1, 11, 2, 11, 3, 11, 4, 11, 5, 11, 6, 11, 7, 11, 8, 11, 9, 11, 10, 11, 1, 12, 5, 12, 7, 12, 11, 12, 1, 13, … — A038568 at the Online Encyclopedia of Integer Sequence

You get the rational fractions ordered by denominator in their simplest form: 1/2, 1/3, 2/3, 1/4, 3/4, 1/5, 2/5, 3/5… There are no pairs like 2/4 and 5/35, because those can be simplified: 2/4 → 1/2; 15/35 → 3/7. You can get the same set of rational fractions by listing every successive pair in this sequence, the Stern-Brocot sequence:

1, 2, 1, 3, 2, 3, 1, 4, 3, 5, 2, 5, 3, 4, 1, 5, 4, 7, 3, 8, 5, 7, 2, 7, 5, 8, 3, 7, 4, 5, 1, 6, 5, 9, 4, 11, 7, 10, 3, 11, 8, 13, 5, 12, 7, 9, 2, 9, 7, 12, 5, 13, 8, 11, 3, 10, 7, 11, 4, 9, 5, 6, 1, 7, 6, 11, 5, 14, 9, 13, 4, 15, 11, 18, 7, 17, 10, 13, 3, 14, 11, 19, 8, 21, 13, 18, 5, 17, 12, 19, … — A002487 at the OEIS

But the fractions don’t come ordered by denominator this time. In fact, they seem to come at random: 1/2, 1/3, 2/3, 1/4, 3/5, 2/5, 3/4, 1/5, 4/7, 3/8, 5/7, 2/7, 5/8… But they’re not random at all. There’s a complicated way of generating them and a simple way. An amazingly simple way, I think:

Moshe Newman proved that the fraction a(n+1)/a(n+2) can be generated from the previous fraction a(n)/a(n+1) = x by 1/(2*floor(x) + 1 – x). The successor function f(x) = 1/(floor(x) + 1 – frac(x)) can also be used. — A002487, “Stern-Brocot Sequence”, at the OEIS

In another form, the Stern-Brocot sequence is generated by what’s called the Calkin-Wilf Tree. Now suppose you use the Stern-Brocot sequence to supply the x co-ordinate of an L-graph whose arms run from 0 to 1. And you use the Calkin-Wilf Tree to supply the y co-ordinate of the L-tree. What do you get? As I described in “I Like Gryke”, you get this fractal:

Limestone fractal

I call it a limestone fractal or pavement fractal or gryke fractal, because it reminds me of the fissured patterns you see in the limestone pavements of the Yorkshire Dales:

Fissured limestone pavement, Yorkshire Dales (Wikipedia)

But what happens when you plot the (x,y) of the Stern-Brocot sequence and the Calkin-Wilf Tree on a circle instead? You get an interestingly distorted limestone fractal:

Limestone fractal on circle

You can also plot the (x,y) around the perimeter of a polygon, then stretch the polygon into a circle. Here’s a square:

Limestone fractal on square

⇓

Limestone square stretched to circle

And here are a pentagon, hexagon, heptagon and octagon — note the interesting perspective effects:

Limestone fractal on pentagon

⇓

Limestone pentagon stretched to circle

Limestone fractal on hexagon

⇓

Limestone hexagon stretched to circle

Limestone fractal on heptagon

⇓

Limestone heptagon stretched to circle

Limestone fractal on octagon

⇓

Limestone octagon stretched to circle

And finally, here are animations of limestone polygons stretching to circles:

Limestone square stretched to circle (animated at EZgif)

Limestone pentagon to circle (animated)

Limestone hexagon to circle (animated)

Limestone heptagon to circle (animated)

Limestone octagon to circle (animated)

Previously Pre-Posted (Please Peruse)

• I Like Gryke — a first look at the limestone fractal

Summer-Time Twos

by in Arithmetic, Base Behavior, Mathematics and tagged , | Leave a comment

I wondered how often the digits of n2 appeared in sum(n1,n2). For example:

17 → 117 = sum(9,17)
20 → 200 = sum(5,20); 204,4; 207,3; 209,2 (c=4)

As I looked at higher n2, I found that the 2-views continued:

63 → 363 = sum(58,63); 638,53; 1638,28; 1763,23; 1863,18 (c=5)
88 → 1288 = sum(73,88); 2788,48; 2881,46; 3388,33; 3880,9; 3888,8 (c=6)
200 → 20009 = sum(14,200); 20022,13; 20034,12; 20045,11; 20055,10; 20064,9;
20072,8; 20079,7; 20085,6; 20090,5; 20094,4; 20097,3; 20099,2 (c=13)
558 → 39558 = sum(483,558); 55833,448; 95583,348; 105558,318; 125580,247; 126558,243; 143558,158; 152558,83; 155583,28; 155808,18; 155825,17; 155841,16; 155856,15; 155870,14; 155883,13; 155895,12 (c=16)
2000 → 2000010 = sum(45,2000); 2000054,44; 2000097,43; 2000139,42; 2000180,41; 2000220,40; 2000259,39; 2000297,38; 2000334,37; 2000370,36; 2000405,35; 2000439,34; 2000472,33; 2000504,32; 2000535,31; 2000565,30;
2000594,29; 2000622,28; 2000649,27; 2000675,26; 2000700,25; 2000724,24; 2000747,23; 2000769,22; 2000790,21; 2000810,20; 2000829,19; 2000847,18; 2000864,17; 2000880,16; 2000895,15; 2000909,14; 2000922,13; 2000934,
12; 2000945,11; 2000955,10; 2000964,9; 2000972,8; 2000979,7; 2000985,6; 2000990,5; 2000994,4; 2000997,3; 2000999,2 (c=44)

But what about other bases?

Base 9

15 in b9 → 115 = sum(5,15) (n=14 in b10) (c=1)
18 in b9 → 118 = sum(11,17); 180,1 (n=17 in b10) (c=2)
20 in b9 → 203 = sum(4,18); 206,3; 208,2 (n=18 in b10) (c=3)
45 in b9 → 445 = sum(32,41); 745,25; 1045,15; 1145,5 (n=41 in b10) (c=4)
55 in b9 → 555 = sum(41,50); 1055,35; 1355,25; 1555,15; 1655,5 (n=50 in b10) (c=5)
65 in b9 → 665 = sum(50,59); 1265,45; 1665,35; 2065,25; 2265,15; 2365,5 (n=59 in b10) (c=6)
75 in b9 → 775 = sum(59,68); 1475,55; 2075,45; 2475,35; 2750,26; 2775,25; 3075,15; 3175,5 (n=68 in b10) (c=8)
85 in b9 → 885 = sum(68,77); 1685,65; 2385,55; 2885,45; 3385,35; 3685,25; 3853,17; 3885,15; 4085,5 (n=77 in b10) (c=9)
200 in b9 → 20003 = sum(13,162); 20016,13; 20028,12; 20040,11; 20050,10; 20058,8; 20066,7; 20073,6; 20078,5; 20083,4; 20086,3; 20088,2 (n=162 in b10) (c=12)
415 in b9 → 13415 = sum(311,338); 25415,345; 36415,315; 41525,302; 46415,275; 55415,245; 63415,215; 64155,212; 70415,175; 75415,145; 80415,115; 83415,75; 85415,45; 86415,15 (n=338 in b10) (c=14)
[…]
2000 in b9 → 2000028 = sum(38,1458); 2000070,41; 2000120,40; 2000158,38; 2000206,37; 2000243,36; 2000278,35; 2000323,34; 2000356,33; 2000388,32; 2000430,31; 2000460,30; 2000488,28; 2000526,27; 2000553,26; 2000578,25; 2000613,24; 2000636,23; 2000658,22; 2000680,21; 2000710,20; 2000728,18; 2000746,17; 2000763,16; 2000778,15; 2000803,14; 2000816,13; 2000828,12; 2000840,11; 2000850,10; 2000858,8; 2000866,7; 2000873,6; 2000878,5; 2000883,4; 2000886,3; 2000888,2 (n=1458 in b10) (c=37)

Base 11

16 in b11 → 116 = sum(6,16) (n=17 in b10) (c=1)
20 in b11 → 201 = sum(5,22); 205,4; 208,3; 20A,2 (n=22 in b10) (c=4)
56 in b11 → 556 = sum(50,61); 956,36; 1156,26; 1356,16; 1456,6 (n=61 in b10) (c=5)
66 in b11 → 666 = sum(61,72); 1066,46; 1466,36; 1669,2A; 1766,26; 1966,16; 1A66,6 (n=72 in b10) (c=7)
86 in b11 → 886 = sum(83,94); 1486,66; 1A86,56; 2486,46; 2886,36; 3086,26; 3286,16; 3386,6 (n=94 in b10) (c=8)
96 in b11 → 996 = sum(94,105); 1696,76; 2296,66; 2896,56; 3296,46; 3696,36; 3996,26; 4096,16; 4196,6 (n=105 in b10) (c=9)
A6 in b11 → AA6 = sum(105,116); 18A6,86; 25A6,76; 31A6,66; 37A6,56; 41A6,46; 45A6,36; 48A6,26; 4AA6,16; 50A6,6 (n=116 in b10) (c=10)
200 in b11 → 1200A = sum(156,242); 20001,15; 20015,14; 20028,13; 2003A,12; 20050,11; 20060,10; 2006A,A; 20078,9; 20085,8; 20091,7; 20097,6; 200A1,5; 200A5,4; 200A8,3; 200AA,2 (n=242 in b10) (c=16)
[…]
A66 in b11 → 1AA66 = sum(1260,1282); A1A66,966; 109A66,946; 182A66,866; 198A66,846; 23A666,786; 253A66,766; 267A66,746; 314A66,666; 326A66,646; 375A66,566; 385A66,546; 416A66,466; 424A66,446; 457A66,366; 463A66,346; 46A666,326; 488A66,266; 492A66,246; 4A6666,186; 4A9A66,166; 501A66,146; 50AA66,66; 510A66,46 (n=1282 in b10) (c=24)
2000 in b11 → 2000005 = sum(52,2662); 2000051,47; 2000097,46; 2000131,45; 2000175,44; 2000208,43; 200024A,42; 2000290,41; 2000320,40; 200035A,3A; 2000398,39; 2000425,38; 2000461,37; 2000497,36; 2000521,35; 2000555,34; 2000588,33; 200060A,32; 2000640,31; 2000670,30; 200069A,2A; 2000718,29; 2000745,28; 2000771,27; 2000797,26; 2000811,25; 2000835,24; 2000858,23; 200087A,22; 20008A0,21; 2000910,20; 200092A,1A; 2000948,19; 2000965,18; 2000981,17; 2000997,16; 2000A01,15; 2000A15,14; 2000A28,13; 2000A3A,12; 2000A50,11; 2000A60,10; 2000A6A,A; 2000A78,9; 2000A85,8; 2000A91,7; 2000A97,6; 2000AA1,5; 2000AA5,4; 2000AA8,3; 2000AAA,2 (n=2662 in b10) (c=51)

Base 3

12 in b3 → 112 = sum(2,12); 120,1 (n=5 in b10) (c=2)
20 in b3 → 120 = sum(4,6); 200,10; 202,2 (n=6 in b10) (c=3)
122 in b3 → 10122 = sum(11,17); 11122,22; 11220,21; 12122,2; 12200,1 (n=17 in b10) (c=5)
1212 in b3 → 121212 = sum(41,50); 1001212,1012; 1101212,212; 1112120,200; 1121212,112; 1201212,12 (n=50 in b10) (c=6)
1222 in b3 → 122222 = sum(44,53); 1101222,1002; 1111222,222; 1112220,221; 1212222,102; 1221222,2; 1222000,1 (n=53 in b10) (c=7)
2000 in b3 → 1112000 = sum(28,54); 1120000,1000; 2000020,21; 2000110,20; 2000122,12; 2000210,11; 2000220,10; 2000222,2 (n=54 in b10) (c=8)
[…]
20000 in b3 → 111120000 = sum(82,162); 111200000,10000; 200000010,111; 200000120,110; 200000222,102; 200001100,101; 200001200,100; 200001222,22; 200002020,21; 200002110,20; 200002122,12; 200002210,11; 200002220,10; 200002222,2 (n=162 in b10) (c=14)

Base 4

13 in b4 → 130 = sum(1,13) (n=7 in b10) (c=1)
20 in b4 → 201 = sum(3,8); 203,2 (n=8 in b10) (c=2)
200 in b4 → 20001 = sum(6,32); 20012,11; 20022,10; 20031,3; 20033,2 (n=32 in b10) (c=5)
2000 in b4 → 2000021 = sum(11,128); 2000103,22; 2000130,21; 2000210,20; 2000223,13; 2000301,12; 2000312,11; 2000322,10; 2000331,3; 2000333,2 (n=128 in b10) (c=10)
20000 in b4 → 200000003 = sum(23,512); 200000121,112; 200000232,111; 200001002,110; 200001111,103; 200001213,102; 200001320,101; 200002020,100; 200002113,33; 200002211,32; 200002302,31; 200002332,30; 200003021,23; 200003103,22; 200003130,21; 200003210,20; 200003223,13; 200003301,12; 200003312,11; 200003322,10; 200003331,3; 200003333,2 (n=512 in b10) (c=22)

Base 8

17 in b8 → 170 = sum(1,17) (n=15 in b10) (c=1)
20 in b8 → 202 = sum(4,16); 205,3; 207,2 (n=16 in b10) (c=3)
200 in b8 → 20011 = sum(11,128); 20023,12; 20034,11; 20044,10; 20053,7; 20061,6; 20066,5; 20072,4; 20075,3; 20077,2 (n=128 in b10) (c=10)
2000 in b8 → 2000020 = sum(32,1024); 2000057,37; 2000115,36; 2000152,35; 2000206,34; 2000241,33; 2000273,32; 2000324,31; 2000354,30; 2000403,27; 2000431,26; 2000456,25; 2000502,24; 2000525,23; 2000547,22; 200057
0,21; 2000610,20; 2000627,17; 2000645,16; 2000662,15; 2000676,14; 2000711,13; 2000723,12; 2000734,11; 2000744,10; 2000753,7; 2000761,6; 2000766,5; 2000772,4; 2000775,3; 2000777,2 (n=1024 in b10) (c=31)

Base 16

1F in b16 → 1F0 = sum(1,1F) (n=31 in b10) (c=1)
20 in b16 → 201 = sum(6,32); 206,5; 20A,4; 20D,3; 20F,2 (n=32 in b10) (c=5)
200 in b16 → 20003 = sum(23,512); 20019,16; 2002E,15; 20042,14; 20055,13; 20067,12; 20078,11; 20088,10; 20097,F; 200A5,E; 200B2,D; 200BE,C; 200C9,B; 200D3,A; 200DC,9; 200E4,8; 200EB,7; 200F1,6; 20
0F6,5; 200FA,4; 200FD,3; 200FF,2 (n=512 in b10) (c=22)
[…]
EE4 in b16 → 42EE4A = sum(961,EE4); 6EE413,16; 6EE428,15; 6EE43C,14; 6EE44F,13; 6EE461,12; 6EE472,11; 6EE482,10; 6EE491,F; 6EE49F,E; 6EE4AC,D; 6EE4B8,C; 6EE4C3,B; 6EE4CD,A; 6EE4D6,9; 6EE4DE,8; 6EE4E5,7; 6EE4EB,6; 6EE4F0,5; 6EE4F4,4; 6EE4F7,3; 6EE4F9,2; 6EE4FA,1 (n=3812 in b10) (c=23)
2000 in b16 → 2000001 = sum(5B,2000); 200005B,5A; 20000B4,59; 200010C,58; 2000163,57; 20001B9,56; 200020E,55; 2000262,54; 20002B5,53; 2000307,52; 2000358,51; 20003A8,50; 20003F7,4F; 2000445,4E; 2000492,4D; 20004DE,4C; 2000529,4B; 2000573,4A; 20005BC,49; 2000604,48; 200064B,47; 2000691,46; 20006D6,45; 200071A,44; 200075D,43; 200079F,42; 20007E0,41; 2000820,40; 200085F,3F; 200089D,3E; 20008DA,3D; 2000916,3C; 2000951,3B; 200098B,3A; 20009C4,39; 20009FC,38; 2000A33,37; 2000A69,36; 2000A9E,35; 2000AD2,34; 2000B05,33; 2000B37,32; 2000B68,31; 2000B98,30; 2000BC7,2F; 2000BF5,2E; 2000C22,2D; 2000C4E,2C; 2000C79,2B; 2000CA3,2A; 2000CCC,29; 2000CF4,28; 2000D1B,27; 2000D41,26; 2000D66,25; 2000D8A,24; 2000DAD,23; 2000DCF,22; 2000DF0,21; 2000E10,20; 2000E2F,1F; 2000E4D,1E; 2000E6A,1D; 2000E86,1C; 2000EA1,1B; 2000EBB,1A; 2000ED4,19; 2000EEC,18; 2000F03,17; 2000F19,16; 2000F2E,15; 2000F42,14; 2000F55,13; 2000F67,12; 2000F78,11; 2000F88,10; 2000F97,F; 2000FA5,E; 2000FB2,D; 2000FBE,C; 2000FC9,B; 2000FD3,A; 2000FDC,9; 2000FE4,8; 2000FEB,7; 2000FF1,6; 2000FF6,5; 2000FFA,4; 2000FFD,3; 2000FFF,2 (n=8192 in b10) (c=90)

Previously Pre-Posted (Please Peruse)

• Summer-Time Hues
• Summer-Climb Views
• Summult-Time Hues

Punctuated Pairimeters

by in Arithmetic, Circles, Geometry, Mathematics and tagged , , , , | Leave a comment

Imagine using the digits of n in two different bases to generate two fractions, a/b and c/d, where a/b < 1 and c/d < 1 (see Appendix for a sample program). Now use the fractions to find a pair of points on the perimeter of a circle, (x1, y1) and (x2, y2), then calculate and mark the midpoint of (x1, y1) and (x2, y2). If the bases have a prime factor in common, pretty patterns will appear from this punctuated pairimetry:

b1 = 2; b2 = 6

b1 = 2; b2 = 10

b1 = 2; b2 = 14

b1 = 4; b2 = 10

b1 = 4; b2 = 20

b1 = 4; b2 = 28

b1 = 6; b2 = 42

b1 = 12; b2 = 39

b1 = 24; b2 = 28

b1 = 28; b2 = 40

b1 = 32; b2 = 36

b1 = 42; b2 = 78

Appendix: Sample Program for Pairimetry

GetXY(xyi)=

fr = 0
recip = 1
bs = base[xyi]
for gi = 1 to di[xyi]
recip = recip/bs
fr += d[xyi,gi] * recip
next gi

x[xyi] = xcenter + sin(pi2 * fr) * radius
y[xyi] = ycenter + cos(pi2 * fr) * radius

endproc

Dinc(i1) =

d[i1,1]++;
if d[i1,1] == base[i1] then

i2 = 1

while d[i1,i2] == base[i1]

d[i1,i2] = 0
i2++;
d[i1,i2]++;

endwhile

if i2 > di[i1] then di[i1] = i2 endif

endif

endproc

Drawfigure =

base = x = y = di = array(2)
d = array(2,100)
radius = 100
pi2 = pi * 2
base[1] = 2
base[2] = 6
di[1] = 1
di[2] = 1

while true

for i = 1 to 2
call Dinc(i)
call GetXY(i)
next i

plot (x[1]+x[2]) / 2, (y[1] + y[2]) / 2

endwhile

endproc

call drawfigure

Summult-Time Hues

by in Arithmetic, Integer Sequences, Mathematics and tagged , , , , | Leave a comment

sum(3,6) = 3 * 6 = 18
• 3 * 2.3 = 2.3^2
sum(15,35) = 15 * 35 = 525
• 3.5 * 5.7 = 3.5^2.7
sum(85,204) = 85 * 204 = 17340
• 5.17 * 2^2.3.17 = 2^2.3.5.17^2
sum(493,1189) = 493 * 1189 = 586177
• 17.29 * 29.41 = 17.29^2.41
sum(2871,6930) = 2871 * 6930 = 19896030
• 3^2.11.29 * 2.3^2.5.7.11 = 2.3^4.5.7.11^2.29
sum(16731,40391) = 16731 * 40391 = 675781821
• 3^2.11.13^2 * 13^2.239 = 3^2.11.13^4.239
[…]

Elsewhere Other-Accessible

1, 18, 525, 17340, 586177, 19896030, 675781821, 22956120408, 779829016225, 26491211221770, 899921240562957, 30570830315362260, 1038508305678375841, 35278711540581704598, 1198437683944896688125, 40711602541832856049200, 1382996048733983114022337 — A011906 at the Online Encyclopedia of Integer Sequences

The Sumber of the B’s

by in Arithmetic, Mathematics and tagged , , , , , , , , , , | Leave a comment

First a bit of a boredom. Then a bit of beauty. These are the triangular numbers, including 666, the Number of the Beast:

1, 3, 6, 10, 15, 21, 28, 36, 45, 55, 66, 78, 91, 105, 120, 136, 153, 171, 190, 210, 231, 253, 276, 300, 325, 351, 378, 406, 435, 465, 496, 528, 561, 595, 630, 666, 703, 741, 780, 820, 861, 903, 946, 990, 1035, 1081, 1128, 1176, 1225, 1275, …

You can createthem as sumbers, that is, as numbers made by summing the whole numbers:

tri(1) = 1 = 1
tri(2) = 3 = 2+1
tri(3) = 6 = 3+2+1
tri(4) = 10 = 4+3+2+1
tri(5) = 15 = 5+4+3+2+1
tri(6) = 21 = 6+5+4+3+2+1
tri(7) = 28 = 7+6+5+4+3+2+1
tri(8) = 36 = 8+7+6+5+4+3+2+1
tri(9) = 45 = 9+8+7+6+5+4+3+2+1
tri(10) = 55 = 10+9+8+7+6+5+4+3+2+1

And here are the square numbers:

1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225, 256, 289, 324, 361, 400, 441, 484, 529, 576, 625, 676, 729, 784, 841, 900, 961, 1024, 1089, 1156, 1225, 1296, 1369, 1444, 1521, 1600, 1681, 1764, 1849, 1936, 2025, 2116, 2209, 2304, 2401, 2500, …

You can create square numbers in various ways. Most obviously, by multiplying each whole number by itself:

sq(1) = 1*1 = 1
sq(2) = 2*2 = 4
sq(3) = 3*3 = 9
sq(4) = 4*4 = 16
sq(5) = 5*5 = 25
sq(6) = 6*6 = 36
sq(7) = 7*7 = 49
sq(8) = 8*8 = 64
sq(9) = 9*9 = 81
sq(10) = 10*10 = 100

Less obviously, by summing consecutive odd numbers:

sq(1) = 1 = 1
sq(2) = 1+3 = 4
sq(3) = 1+3+5 = 9
sq(4) = 1+3+5+7 = 16
sq(5) = 1+3+5+7+9 = 25
sq(6) = 1+3+5+7+9+11 = 36
sq(7) = 1+3+5+7+9+11+13 = 49
sq(8) = 1+3+5+7+9+11+13+15 = 64
sq(9) = 1+3+5+7+9+11+13+15+17 = 81
sq(10) = 1+3+5+7+9+11+13+15+17+19 = 100

And by summing pairs of consecutive triangular numbers (note that tri(0) = 0):

sq(1) = tri(0) + tri(1) = 0 + 1 = 1
sq(2) = tri(1) + tri(2) = 1 + 3 = 4
sq(3) = tri(2) + tri(3) = 3 + 6 = 9
sq(4) = tri(3) + tri(4) = 6 + 10 = 16
sq(5) = tri(4) + tri(5) = 10 + 15 = 25
sq(6) = tri(5) + tri(6) = 15 + 21 = 36
sq(7) = tri(6) + tri(7) = 21 + 28 = 49
sq(8) = tri(7) + tri(8) = 28 + 36 = 64
sq(9) = tri(8) + tri(9) = 36 + 45 = 81
sq(10) = tri(9) + tri(10) = 45 + 55 = 100

But sometimes squares are the sum of two triangular numbers that aren’t consecutive:

sq(4) = tri(1) + tri(5) = 1+15 = 16
sq(9) = tri(2) + tri(12) = 3+78 = 81
sq(16) = tri(2) + tri(22) = 3+253 = 256
sq(52) = tri(2) + tri(73) = 3+2701 = 2704
sq(14) = tri(3) + tri(19) = 6+190 = 196
sq(21) = tri(3) + tri(29) = 6+435 = 441
sq(44) = tri(9) + tri(61) = 45+1891 = 1936
sq(51) = tri(9) + tri(71) = 45+2556 = 2601
sq(49) = tri(10) + tri(68) = 55+2346 = 2401
sq(56) = tri(10) + tri(78) = 55+3081 = 3136
sq(16) = tri(11) + tri(19) = 66+190 = 256
sq(38) = tri(11) + tri(52) = 66+1378 = 1444
sq(54) = tri(11) + tri(75) = 66+2850 = 2916
sq(87) = tri(47) + tri(113) = 1128+6441 = 7569
sq(77) = tri(48) + tri(97) = 1176+4753 = 5929
sq(121) = tri(64) + tri(158) = 2080+12561 = 14641
sq(141) = tri(96) + tri(174) = 4656+15225 = 19881
sq(121) = tri(100) + tri(138) = 5050+9591 = 14641

Here’s a graph of squares that are the sum of any two triangular numbers, that is, is_square(tri(k1)+tri(k2)). The x axis is 1..k1 and the y axis is 1..k2, so the graph is symmetrical:

tri(k1) + tri(k2) = square(k3)

The (double) line at 45° represents squares that are the sum of consecutive triangulars. Other lines represent similarly regular patterns. Now for a bit of beauty. Things get more visually interesting when you test for squares that are the sums of any integer and a triangular number:

k1 + tri(k2) = square(k3)

The curves are optical oddities: where do they begin and end? The upper ones become lost to the eye in the lower ones. And vice versa. But you can force your eye to trace them further that it wants to.

Now try sums of integers and other polygonal numbers:

k1 + tri(k2) = pentagonal(k3)

k1 + square(k2) = pentagonal(k3)

k1 + pentagonal(k2) = square(k3)

k1 + hexagonal(k2) = pentagonal(k3)

And try other number sequences, like multiples of 4 with polygonals:

k1*4 + pentagonal(k2) = tri(k3)

k1*4 + square(k2) = tri(k3)

k1*4 + heptagonal(k2) = tri(k3)

And primes with polygonals:

tri(k1) + prime(k2) = tri(k3)

prime(k1) + tri(k2) = square(k3)

prime(k1) + octagonal(k2) = square(k3)

prime(k1) + pentagonal(k2) = square(k3)

prime(k1) + square(k2) = decagonal(k3)

prime(k1) + tri(k2) = hendecagonal(k3)

Partitional Pulchritude

by in Arithmetic, Geometry, Integer Sequences, Mathematics and tagged , , , , , , | Leave a comment

If you want a good example of how, in math, something very simple can quickly get very deep, just look at partitions. Here are the partitions of 1 to 5, that is, the ways 1 to 5 can be expressed as a sum of integers smaller than or equal to themselves:

1 = 1

numbpart(1) = 1

2 = 2
1 + 1 = 2

numbpart(2) = 2

3 = 3
1 + 2 = 3
1 + 1 + 1 = 3

numbpart(3) = 3

4 = 4
1 + 3 = 4
2 + 2 = 4
1 + 1 + 2 = 4
1 + 1 + 1 + 1 = 4

numbpart(4) = 5

5 = 5
1 + 4 = 5
2 + 3 = 5
1 + 1 + 3 = 5
1 + 2 + 2 = 5
1 + 1 + 1 + 2 = 5
1 + 1 + 1 + 1 + 1 = 5

numbpart(5) = 7

It’s very easy to understand the concept of partitions, but very difficult to understand how partitions behave. For example, here is numbpart(n), the count of partitions for 1, 2, 3,…

1, 2, 3, 5, 7, 11, 15, 22, 30, 42, 56, 77, 101, 135, 176, 231, 297, 385, 490, 627, 792, 1002, 1255, 1575, 1958, 2436, 3010, 3718, 4565, 5604, 6842, 8349, 10143, 12310, 14883, 17977, 21637, 26015, 31185, 37338, 44583, 53174, 63261, 75175, 89134, 105558, 124754, 147273, 173525, 204226, … A000041 at the Online Encyclopedia of Integer Sequences, “a(n) is the number of partitions of n (the partition numbers)”

What’s the formula for numbpart(n)? That’s a tricky question. And what’s the formula for the curves produced by counting the various lengths of partitions(n)? That’s another tricky question, but one thing is easy to see. As n gets bigger, the graph of countlen(partitions(n)) acquires a strange, lopsided beauty. Here are the partitions of 8, with the count of how many partitions of a particular length there are:

8 = 8 (1 partition of length 1)
1 + 7 = 8
2 + 6 = 8
3 + 5 = 8
4 + 4 = 8 (4 partitions of length 2)
1 + 1 + 6 = 8
1 + 2 + 5 = 8
1 + 3 + 4 = 8
2 + 2 + 4 = 8
2 + 3 + 3 = 8 (5 of length 3)
1 + 1 + 1 + 5 = 8
1 + 1 + 2 + 4 = 8
1 + 1 + 3 + 3 = 8
1 + 2 + 2 + 3 = 8
2 + 2 + 2 + 2 = 8 (5 of length 4)
1 + 1 + 1 + 1 + 4 = 8
1 + 1 + 1 + 2 + 3 = 8
1 + 1 + 2 + 2 + 2 = 8 (3 of length 5)
1 + 1 + 1 + 1 + 1 + 3 = 8
1 + 1 + 1 + 1 + 2 + 2 = 8 (2 of length 6)
1 + 1 + 1 + 1 + 1 + 1 + 2 = 8 (1 of length 7)
1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 = 8 (1 of length 8)

When counts like that are shown as a graph, the graphs look like this (maximum counts are normalized to the same height):


graph of countlen(partitions(2))


countlen(partitions(3))


countlen(partitions(4))


countlen(partitions(5))


countlen(partitions(6))


countlen(partitions(7))


countlen(partitions(8))


countlen(partitions(9))


countlen(partitions(10))


countlen(partitions(15))


countlen(partitions(20))


countlen(partitions(30))


countlen(partitions(40))


countlen(partitions(50))


countlen(partitions(60))


countlen(partitions(70))


countlen(partitions(80))


countlen(partitions(90))


countlen(partitions(100))


Animated gif of partlen graphs (courtesy EZgif)

The graphs have a long, low right tail because the counts rise to great heights very quick, then fall away again, as you can see with partitions(100):

1 = count(partitions(10),len=1)
50 = count(partitions(10),len=2)
833 = count(partitions(10),len=3)
7153 = count(partitions(10),len=4)
38225 = count(partitions(10),len=5)
143247 = count(partitions(10),len=6)

[…]

10643083 = count(partitions(10),len=16)
11022546 = count(partitions(10),len=17)
11087828 = count(partitions(10),len=18)
10885999 = count(partitions(10),len=19)
10474462 = count(partitions(10),len=20)

[…]

30 = count(partitions(10),len=91)
22 = count(partitions(10),len=92)
15 = count(partitions(10),len=93)
11 = count(partitions(10),len=94)
7 = count(partitions(10),len=95)
5 = count(partitions(10),len=96)
3 = count(partitions(10),len=97)
2 = count(partitions(10),len=98)
1 = count(partitions(10),len=99)
1 = count(partitions(10),len=100)

Summer-Time Hues

by in Arithmetic, Mathematics | Leave a comment

sum(4,17) = 147 = 4 + 5 + 6 + 7 + 8 + 9 + 10 + 11 + 12 + 13 + 14 + 15 + 16 + 17
sum(1,20) = 210
sum(19,59) = 1599
sum(22,77) = 2772
sum(20,156) = 12056
sum(34,167) = 13467
sum(23,211) = 22113
sum(79,227) = 22797
sum(84,229) = 22849
sum(61,236) = 26136
sum(199,599) = 159999
sum(203,771) = 277103
sum(222,777) = 277722
sum(266,778) = 267786
sum(277,797) = 279777
sum(145,1520) = 1145520
sum(117,1522) = 1152217
sum(149,1525) = 1152549
sum(167,1527) = 1152767
sum(208,1568) = 1208568
sum(334,1667) = 1334667
sum(540,1836) = 1540836
sum(315,1990) = 1931590
sum(414,2062) = 2041462
sum(418,2063) = 2041863
sum(158,2083) = 2158083
sum(244,2132) = 2244132
sum(554,2139) = 2135549
sum(902,2347) = 2349027
sum(883,2558) = 2883558
sum(989,2637) = 2989637
sum(436,2963) = 4296336
sum(503,3303) = 5330303
sum(626,3586) = 6235866
sum(816,4183) = 8418316
sum(1075,4700) = 10470075
sum(1117,4922) = 11492217
sum(1306,5273) = 13052736
sum(1377,5382) = 13538277
sum(1420,5579) = 14557920
sum(1999,5999) = 15999999
sum(2727,7272) = 22727727
sum(2516,7528) = 25175286
sum(2625,7774) = 26777425
sum(2222,7777) = 27777222
sum(3765,9490) = 37949065
sum(535,10319) = 53103195
sum(1101,14973) = 111497301
sum(2088,15688) = 120885688
sum(3334,16667) = 133346667
sum(2603,19798) = 192603798
sum(3093,19893) = 193093893
sum(1162,20039) = 200116239
sum(1415,20095) = 200914155
sum(1563,20118) = 201156318
sum(2707,20294) = 202270794
sum(2518,20318) = 203251818
sum(2608,20333) = 203326083
sum(2895,20370) = 203289570
sum(3424,20552) = 205342452
sum(4255,20855) = 208425555
sum(4571,20971) = 209457171
sum(4613,21028) = 210461328
sum(4742,21259) = 214742259
sum(6318,21798) = 217631898
sum(6498,21943) = 219649843
sum(7080,22305) = 223708005
sum(7243,22358) = 223724358
sum(6833,22368) = 226833368
sum(7128,22473) = 227128473
sum(4523,22603) = 245232603
sum(4978,22898) = 249782898
sum(8339,23019) = 230183399
sum(8610,23191) = 231861091
sum(6013,23588) = 260133588
sum(9252,23652) = 236925252
sum(6488,23913) = 264883913
sum(8379,25254) = 283795254
sum(4012,28667) = 402866712
sum(4922,31762) = 492317622
sum(4998,31801) = 493180198
sum(5200,32675) = 520326750
sum(7707,40092) = 774009207
sum(7868,40431) = 786404318
sum(9325,44450) = 944450325
sum(11047,48287) = 1104828747
sum(14699,56100) = 1465610099
sum(16235,59860) = 1659860235
sum(19999,59999) = 1599999999
sum(17264,61239) = 1726123964
sum(17405,61605) = 1746160505
sum(18457,63782) = 1863782457
sum(25016,75028) = 2501750286
sum(28022,79942) = 2802799422
sum(37060,93740) = 3706937400
sum(7567,119567) = 7119567567
sum(9638,135513) = 9135513638
sum(15392,152607) = 11526075392
sum(17744,152880) = 11528807744
sum(12012,156387) = 12156387012
sum(20888,156888) = 12088856888
sum(30663,164538) = 13066364538
sum(33334,166667) = 13333466667
sum(36038,168838) = 13603868838

Factory Façades

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Practically speaking, I’d never heard of them. Practical numbers, that is. They’re defined like this at the Online Encyclopedia of Integer Sequences:

A005153 Practical numbers: positive integers m such that every k <= sigma(m) is a sum of distinct divisors of m. Also called panarithmic numbers. […] Equivalently, positive integers m such that every number k <= m is a sum of distinct divisors of m. — A005153 at OEIS

In other words, if you take, say, divisors(12) = 1, 2, 3, 4, 6, you can find partial sums of those divisors that equal every number from 1 to 16, where 16 = 1+2+3+4+6. Here are all those sums, with c as the count of divisor-sums equalling a particular k (to simplify things, I’m excluding 12 as a divisor of 12):

1, 2, 3, 4, 6 = divisors(12)

01 = 1 (c=1)
02 = 2 (c=1)
03 = 1 + 2 = 3 (c=2)
04 = 1 + 3 = 4 (c=2)
05 = 2 + 3 = 1 + 4 (c=2)
06 = 1 + 2 + 3 = 2 + 4 = 6 (c=3)
07 = 1 + 2 + 4 = 3 + 4 = 1 + 6 (c=3)
08 = 1 + 3 + 4 = 2 + 6 (c=2)
09 = 2 + 3 + 4 = 1 + 2 + 6 = 3 + 6 (c=3)
10 = 1 + 2 + 3 + 4 = 1 + 3 + 6 = 4 + 6 (c=3)
11 = 2 + 3 + 6 = 1 + 4 + 6 (c=2)
12 = 1 + 2 + 3 + 6 = 2 + 4 + 6 (c=2)
13 = 1 + 2 + 4 + 6 = 3 + 4 + 6 (c=2)
14 = 1 + 3 + 4 + 6 (c=1)
15 = 2 + 3 + 4 + 6 (c=1)
16 = 1 + 2 + 3 + 4 + 6 (c=1)

Learning about practical numbers inspired me to look at the graphs of the count of the divisor-sums for 12. If you include count(0) = 1 (there is one way of choosing divisors of 12 to equal 0, namely, by choosing none of the divisors), the graph looks like this:

counts of divisorsum(12) = k, where 12 = 2^2 * 3 → 1, 2, 3, 4, 6

Here are some more graphs for partialsumcount(n), adjusted for a standardized y-max. They remind me variously of skyscrapers, pyramids, stupas, factories and factory façades, forts bristling with radar antennae, and the Houses of Parliament. All in an art-deco style:

18 = 2 * 3^2 → 1, 2, 3, 6, 9

24 = 2^3 * 3 → 1, 2, 3, 4, 6, 8, 12

30 = 2 * 3 * 5 → 1, 2, 3, 5, 6, 10, 15

36 = 2^2 * 3^2 → 1, 2, 3, 4, 6, 9, 12, 18

48 = 2^4 * 3 → 1, 2, 3, 4, 6, 8, 12, 16, 24

54 = 2 * 3^3 → 1, 2, 3, 6, 9, 18, 27

60 = 2^2 * 3 * 5 → 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30

72 = 2^3 * 3^2 → 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36

88 = 2^3 * 11 → 1, 2, 4, 8, 11, 22, 44, 88

96 = 2^5 * 3 → 1, 2, 3, 4, 6, 8, 12, 16, 24, 32, 48

100 = 2^2 * 5^2 → 1, 2, 4, 5, 10, 20, 25, 50

108 = 2^2 * 3^3 → 1, 2, 3, 4, 6, 9, 12, 18, 27, 36, 54

120 = 2^3 * 3 * 5 → 1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 20, 24, 30, 40, 60

126 = 2 * 3^2 * 7 → 1, 2, 3, 6, 7, 9, 14, 18, 21, 42, 63

162 = 2 * 3^4 → 1, 2, 3, 6, 9, 18, 27, 54, 81

220 = 2^2 * 5 * 11 → 1, 2, 4, 5, 10, 11, 20, 22, 44, 55, 110

And what about im-practical numbers, where the partial sums of divisors(m) don’t equal every number 1..sigma(m)? There are interesting fractal patterns to be uncovered there, as you can see from the graph for 190 (because all divsumcount(k) = 1, the graph looks like a bar-code):

190 = 2 * 5 * 19 → 1, 2, 5, 10, 19, 38, 95

Fertile Fractions

by in Arithmetic, Egyptian Fractions, Mathematics, Reciprocals and tagged , , | Leave a comment

Here’s the simplest possible Egyptian fraction summing to 1:

1 = 1/2 + 1/3 + 1/6 = egypt(2,3,6)

But how many times does 1 = egypt()? Infinitely often, as is very easy to prove. Take this equation:

1/6 – 1/7 = 1/42

For any 1/n, 1/n – 1/(n+1) = 1/(n*(n+1)) = 1/(n^2 + n). In the case of 1/6, the formula means that you can re-write egypt(2,3,6) like this:

1 = egypt(2,3,7,42) = 1/2 + 1/3 + 1/7 + 1/42

Now try these equations:

1/6 – 1/8 = 1/24
1/6 – 1/9 = 1/18
1/6 – 1/10 = 1/15

Which lead to these re-writes of egypt(2,3,6):

1 = egypt(2,3,8,24)
1 = egypt(2,3,9,18)
1 = egypt(2,3,10,15)

Alternatively, you can expand the 1/3 of egypt(2,3,6):

1/3 – 1/4 = 1/12

Therefore:

1 = egypt(2,4,6,12)

And the 1/12 opens all these possibilities:

1/12 – 1/13 = 1/156 → 1 = egypt(2,4,6,13,156)
1/12 – 1/14 = 1/84 → 1 = egypt(2,4,6,14,84)
1/12 – 1/15 = 1/60 → 1 = egypt(2,4,6,15,60)
1/12 – 1/16 = 1/48 → 1 = egypt(2,4,6,16,48)
1/12 – 1/18 = 1/36 → 1 = egypt(2,4,6,18,36)
1/12 – 1/20 = 1/30 → 1 = egypt(2,4,6,20,30)
1/12 – 1/21 = 1/28 → 1 = egypt(2,4,6,21,28)

So you can expand an Egyptian fraction for ever. If you stick to expanding the 1/6 to 1/7 + 1/42, then the 1/42 to 1/43 + 1/1806 and so on, you get this:

1 = egypt(2,3,6)
1 = egypt(2,3,7,42)
1 = egypt(2,3,7,43,1806)
1 = egypt(2,3,7,43,1807,3263442)
1 = egypt(2,3,7,43,1807,3263443,10650056950806)
1 = egypt(2,3,7,43,1807,3263443,10650056950807,113423713055421844361000442)
1 = egypt(2,3,7,43,1807,3263443,10650056950807,113423713055421844361000443,12864938683278671740537145998360961546653259485195806)
1 = egypt(2,3,7,43,1807,3263443,10650056950807,113423713055421844361000443,12864938683278671740537145998360961546653259485195807,165506647324519964198468195444439180017513152706377497841851388766535868639572406808911988131737645185442)
[…]

Elsewhere Other-Accessible…

• A000058, Sylvester’s Sequence, at the Online Encyclopedia of Integer Sequences, with more details on the numbers above