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):**