East, west, home’s best. And for human beings, base-10 is a kind of home. We have ten fingers and we use ten digits. Base-10 comes naturally to us: it feels like home. So it’s disappointing that there is no number in base-10 that is equal to the sum of the squares of its digits (apart from the trivial 0^2 = 0 and 1^2 = 1). Base-9 and base-11 do better:

41 = 45_[b=9] = 4^2 + 5^2 = 16 + 25 = 41
50 = 55_[b=9] = 5^2 + 5^2 = 25 + 25 = 50

61 = 56_[b=11] = 5^2 + 6^2 = 25 + 36 = 61
72 = 66_[b=11] = 6^2 + 6^2 = 36 + 36 = 72

Base-47 does better still, with fourteen 2-sumbers. And base-10 does have 3-sumbers, or numbers equal to the sum of the cubes of their digits:

153 = 1^3 + 5^3 + 3^3 = 1 + 125 + 27 = 153
370 = 3^3 + 7^3 + 0^3 = 27 + 343 + 0 = 370
371 = 3^3 + 7^3 + 1^3 = 27 + 343 + 1 = 371
407 = 4^3 + 0^3 + 7^3 = 64 + 0 + 343 = 407

But base-10 disappoints again when it comes to prumbers, or prime sumbers, or numbers that are equal to the sum of the primes whose indices are equal to the digits of the number. The index of a prime number is its position in the list of primes. Here are the first nine primes and their indices (with 0 as a pseudo-prime at position 0):

prime(0) = 0
prime(1) = 2
prime(2) = 3
prime(3) = 5
prime(4) = 7
prime(5) = 11
prime(6) = 13
prime(7) = 17
prime(8) = 19
prime(9) = 23

So the prumber, or prime-sumber, of 1 = prime(1) = 2. The prumber of 104 = prime(1) + prime(0) + prime(4) = 2 + 0 + 7 = 9. The prumber of 186 = 2 + 19 + 13 = 34. But no number in base-10 is equal to its prime sumber. Base-2 and base-3 do better:

Base-2 has 1 prumber:

2 = 10_[b=2] = 2 + 0 = 2

Base-3 has 2 prumbers:

4 = 11_[b=3] = 2 + 2 = 4
5 = 12_[b=3] = 2 + 3 = 5

But prumbers are rare. The next record is set by base-127, with 4 prumbers:

165 = 1[38]_[b=127] = 2 + 163 = 165
320 = 2[66]_[b=127] = 3 + 317 = 320
472 = 3[91]_[b=127] = 5 + 467 = 472
620 = 4[112]_[b=127] = 7 + 613 = 620

Base-479 has 4 prumbers:

1702 = 3[265] = 5 + 1697 = 1702
2250 = 4[334] = 7 + 2243 = 2250
2800 = 5[405] = 11 + 2789 = 2800
3344 = 6[470] = 13 + 3331 = 3344

Base-637 has 4 prumbers:

1514 = 2[240] = 3 + 1511 = 1514
2244 = 3[333] = 5 + 2239 = 2244
2976 = 4[428] = 7 + 2969 = 2976
4422 = 6[600] = 13 + 4409 = 4422

Base-831 has 4 prumbers:

999 = 1[168] = 2 + 997 = 999
2914 = 3[421] = 5 + 2909 = 2914
3858 = 4[534] = 7 + 3851 = 3858
4798 = 5[643] = 11 + 4787 = 4798

Base-876 has 4 prumbers:

1053 = 1[177] = 2 + 1051 = 1053
3066 = 3[438] = 5 + 3061 = 3066
4064 = 4[560] = 7 + 4057 = 4064
6042 = 6[786] = 13 + 6029 = 6042

Previously pre-posted (please peruse):

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