# Lord, What Fuels These Portals Be!

Midnight, one more night without sleeping.
Watching ’til that morning comes creeping.
Green Door: what’s that secret you’re keeping?

There’s an old piano and they play it hot behind the green door!
Don’t know what they’re doing but they laugh a lot behind the green door.
Wish they’d let me in so I could find out what’s behind the green door.

Knocked once, tried to tell ’em I’d been there.
Door slammed — hospitality’s thin there.
Wondering just what’s going on in there.

Saw an eyeball peepin’ through a smoky cloud behind the green door.
When I said “Joe sent me” someone laughed out loud behind the green door.
All I want to do is join the happy crowd behind the green door.

Otra noche mas que no duermo.
Otra noche mas que se pierde.
¿Que habrá tras esa puerta verde?
Suena alegremente un piano viejo
tras la puerta verde.

Todos ríen y no se que pasa
tras la puerta verde
No descansaré hasta saber que hay
tras la puerta verde.

Toqué, y cuando contestaron
dije ¡Ah! que a mí me llamaron.
Risas, y enseguida me echaron.

Sólo pude ver que mucha gente allí se divertía,
y entre tanto humo todo allí se confundía.
Yo quisiera estar al otro lado de la puerta verde.

Otra noche mas que no duermo.
Otra noche mas que se pierde.
¿Que habrá tras esa puerta verde?
¿Que habrá tras esa puerta verde?
¿Que habrá?

Elsewhere Other-Accessible…

“The Green Door” (1956), music by Bob “Hutch” Davie and lyrics by Marvin J. Moore

# Primal Stream

It’s obvious when you think about: an even number can never be the sum of two consecutive integers. Conversely, an odd number (except 1) is always the sum of two consecutive integers: 3 = 1 + 2; 5 = 2 + 3; 7 = 3 + 4; 9 = 4 + 5; and so on. The sum of three consecutive integers can be either odd or even: 6 = 1 + 2 + 3; 9 = 2 + 3 + 4. The sum of four consecutive integers must always be even: 1 + 2 + 3 + 4 = 10; 2 + 3 + 4 + 5 = 14. And so on.

But notice that 9 is the sum of consecutive integers in two different ways: 9 = 4 + 5 = 2 + 3 + 4. Having spotted that, I decided to look for numbers that were the sums of consecutive integers in the most different ways. These are the first few:

3 = 1 + 2 (number of sums = 1)
9 = 2 + 3 + 4 = 4 + 5 (s = 2)
15 = 1 + 2 + 3 + 4 + 5 = 4 + 5 + 6 = 8 + 7 = (s = 3)
45 (s = 5)
105 (s = 7)
225 (s = 8)
315 (s = 11)
945 (s = 15)
1575 (s = 17)
2835 (s = 19)
3465 (s = 23)
10395 (s = 31)

It was interesting that the number of different consecutive-integer sums for n was most often a prime number. Next I looked for the sequence at the Online Encyclopedia of Integer Sequences and discovered something that I hadn’t suspected:

A053624 Highly composite odd numbers: where d(n) increases to a record.

1, 3, 9, 15, 45, 105, 225, 315, 945, 1575, 2835, 3465, 10395, 17325, 31185, 45045, 121275, 135135, 225225, 405405, 675675, 1576575, 2027025, 2297295, 3828825, 6891885, 11486475, 26801775, 34459425, 43648605, 72747675, 130945815 — A053624 at OEIS

The notes add that the sequence is “Also least number k such that the number of partitions of k into consecutive integers is a record. For example, 45 = 22+23 = 14+15+16 = 7+8+9+10+11 = 5+6+7+8+9+10 = 1+2+3+4+5+6+7+8+9, six such partitions, but all smaller terms have fewer such partitions (15 has four).” When you don’t count the number n itself as a partition of n, you get 3 partitions for 15, i.e. consecutive integers sum to 15 in 3 different ways, so s = 3. I looked at more values for s and found that the stream of primes continued to flow:

3 → s = 1
9 = 3^2 → s = 2 (prime)
15 = 3 * 5 → s = 3 (prime)
45 = 3^2 * 5 → s = 5 (prime)
105 = 3 * 5 * 7 → s = 7 (prime)
225 = 3^2 * 5^2 → s = 8 = 2^3
315 = 3^2 * 5 * 7 → s = 11 (prime)
945 = 3^3 * 5 * 7 → s = 15 = 3 * 5
1575 = 3^2 * 5^2 * 7 → s = 17 (prime)
2835 = 3^4 * 5 * 7 → s = 19 (prime)
3465 = 3^2 * 5 * 7 * 11 → s = 23 (prime)
10395 = 3^3 * 5 * 7 * 11 → s = 31 (prime)
17325 = 3^2 * 5^2 * 7 * 11 → s = 35 = 5 * 7
31185 = 3^4 * 5 * 7 * 11 → s = 39 = 3 * 13
45045 = 3^2 * 5 * 7 * 11 * 13 → s = 47 (prime)
121275 = 3^2 * 5^2 * 7^2 * 11 → s = 53 (prime)
135135 = 3^3 * 5 * 7 * 11 * 13 → s = 63 = 3^2 * 7
225225 = 3^2 * 5^2 * 7 * 11 * 13 → s = 71 (prime)
405405 = 3^4 * 5 * 7 * 11 * 13 → s = 79 (prime)
675675 = 3^3 * 5^2 * 7 * 11 * 13 → s = 95 = 5 * 19
1576575 = 3^2 * 5^2 * 7^2 * 11 * 13 → s = 107 (prime)
2027025 = 3^4 * 5^2 * 7 * 11 * 13 → s = 119 = 7 * 17
2297295 = 3^3 * 5 * 7 * 11 * 13 * 17 → s = 127 (prime)
3828825 = 3^2 * 5^2 * 7 * 11 * 13 * 17 → s = 143 = 11 * 13
6891885 = 3^4 * 5 * 7 * 11 * 13 * 17 → s = 159 = 3 * 53
11486475 = 3^3 * 5^2 * 7 * 11 * 13 * 17 → s = 191 (prime)
26801775 = 3^2 * 5^2 * 7^2 * 11 * 13 * 17 → s = 215 = 5 * 43
34459425 = 3^4 * 5^2 * 7 * 11 * 13 * 17 → s = 239 (prime)
43648605 = 3^3 * 5 * 7 * 11 * 13 * 17 * 19 → s = 255 = 3 * 5 * 17
72747675 = 3^2 * 5^2 * 7 * 11 * 13 * 17 * 19 → s = 287 = 7 * 41
130945815 = 3^4 * 5 * 7 * 11 * 13 * 17 * 19 → s = 319 = 11 * 29

I can’t spot any way of predicting when n will yield a primal s, but I like the way that a simple question took an unexpected turn. When a number sets a record for the number of different ways it can be the sum of consecutive integers, that number will also be a highly composite odd number.

# Tri + Eye = Troculus

Troculus, a fractal Lovecraftian entity created by dividing-and-discarding parts of a triangle

Troculus converted into a circle

Troculus switching between forms (animated gif)

Elsewhere Other-Accessible…

Circus Trix — how to create Troculus & Co.

# Performativizing Papyrocentricity #74

Papyrocentric Performativity Presents…

A Big Book about BooksThe Penguin Classics Book, Henry Eliot (Penguin 2018)

Wrecks & Drugs & Rock & RollBodies: Life and Death in Music, Ian Winwood (Faber 2022)

In the Bland of the BlindAn Unexplained Death: The True Story of a Body at the Belvedere, Mikita Brottman (Canongate 2018)

Hu Thru MuThe Musical Human: A History of Life on Earth, Michael Spitzer (Bloomsbury 2021)

A Bit of EngLitThe Power of Delight: A Lifetime in Literature: Essays 1962-2002, John Bayley (Duckworth 2005)

Chrome TomeThe Secret Lives of Colour, Kassia St Clair (John Murray 2018)

Cannonball Corpse – AC/DC: The Story of the Original Monsters of Rock, Jerry Ewing (Carlton Books 2015)

Chimpathy for the Devil?Oasis: Supersonic: The Complete, Authorised and Uncut Interviews, curated by Simon Halfon (Nemperor 2021)

D for Deviant, K for Korpse…Doktor Deviant’s Diary of Depravity: Kandid Konfessions of a Kompulsive Korpse-Kopulator, ed. Dr David Kerekes and Samuel P. Salatta (Visceral Visions 2022)

Or Read a Review at Random: RaRaR

# You Sixy Beast

666 is the Number of the Beast. But it’s much more than that. After all, it’s a number, so it has mathematical properties (everything has mathematical properties, but it’s a sine-qua-non of numbers). For example, 666 is a palindromic number, reading the same forwards and backwards. And it’s a repdigit, consisting of a single repeated digit. Now try answering this question: how many pebbles are there in this triangle?

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Counting the pebbles one by one would take a long time, but there’s a short-cut. Each line of the triangle after the first is one pebble longer than the previous line. There are 36 lines and therefore 36 pebbles in the final line. So the full number of pebbles = 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 + 11 + 12 + 13 + 14 + 15 + 16 + 17 + 18 + 19 + 20 + 21 + 22 + 23 + 24 + 25 + 26 + 27 + 28 + 29 + 30 + 31 + 32 + 33 + 34 + 35 + 36. And there’s an easy formula for that sum: (36^2 + 36) / 2 = (1296 + 36) / 2 = 1332 / 2 = 666.

So 666 is the 36th triangular number:

1 = 1
1+2 = 3
1+2+3 = 6
1+2+3+4 = 10
1+2+3+4+5 = 15
1+2+3+4+5+6 = 21
1+2+3+4+5+6+7 = 28
1+2+3+4+5+6+7+8 = 36
1+2+3+4+5+6+7+8+9 = 45
1+2+3+4+5+6+7+8+9+10 = 55
[...]
1+2+3+4+5+6+7+8+9+10+11+12+13+14+15+16+17+18+19+20+21+22+23+24+25+26+27+28+29+30+31+32+33+34+35+36 = 666

But what’s tri(666), the 666th triangular number? By the formula above, it equals (666^2 + 666) / 2 = (443556 + 666) / 2 = 444222 / 2 = 222111. But recall something else from above: tri(6) = 1+2+3+4+5+6 = 21. Is it a coincidence that tri(6) = 21 and tri(666) = 222111? No, it isn’t:

tri(6) = 21 = (6^2 + 6) / 2 = (36 + 6) / 2 = 42 / 2
tri(66) = 2211 = (66^2 + 66) / 2 = (4356 + 66) / 2 = 4422 / 2
tri(666) = 222111 = (666^2 + 666) / 2 = (443556 + 666) / 2 = 444222 / 2
tri(6666) = 22221111
tri(66666) = 2222211111
tri(666666) = 222222111111
tri(6666666) = 22222221111111
tri(66666666) = 2222222211111111
tri(666666666) = 222222222111111111
tri(6666666666) = 22222222221111111111
tri(66666666666) = 2222222222211111111111
tri(666666666666) = 222222222222111111111111
tri(6666666666666) = 22222222222221111111111111
tri(66666666666666) = 2222222222222211111111111111
tri(666666666666666) = 222222222222222111111111111111

So we’ve looked at tri(36) = 666 and tri(666) = 222111. Let’s go a step further: tri(222111) = 24666759216. So 666 appears again. And the sixiness carries on here:

tri(36) = 666
tri(3366) = 5666661
tri(333666) = 55666666611
tri(33336666) = 555666666666111
tri(3333366666) = 5555666666666661111
tri(333333666666) = 55555666666666666611111
tri(33333336666666) = 555555666666666666666111111
tri(3333333366666666) = 5555555666666666666666661111111
tri(333333333666666666) = 55555555666666666666666666611111111
tri(33333333336666666666) = 555555555666666666666666666666111111111
tri(3333333333366666666666) = 5555555555666666666666666666666661111111111
tri(333333333333666666666666) = 55555555555666666666666666666666666611111111111
tri(33333333333336666666666666) = 555555555555666666666666666666666666666111111111111
tri(3333333333333366666666666666) = 5555555555555666666666666666666666666666661111111111111
tri(333333333333333666666666666666) = 55555555555555666666666666666666666666666666611111111111111