89 is special because it’s a prime number, divisible by only itself and 1. It’s also a sum of powers in a special way: 89 = 8^1 + 9^2. In base ten, no other two-digit number is equal to its own ascending power-sum like that. But the same pattern appears in these three-digit numbers, as the powers climb with the digits:

135 = 1^1 + 3^2 + 5^3 = 1 + 9 + 125 = 135

175 = 1^1 + 7^2 + 5^3 = 1 + 49 + 125 = 175

518 = 5^1 + 1^2 + 8^3 = 5 + 1 + 512 = 518

598 = 5^1 + 9^2 + 8^3 = 5 + 81 + 512 = 598

And in these four-digit numbers:

1306 = 1^1 + 3^2 + 0^3 + 6^4 = 1 + 9 + 0 + 1296 = 1306

1676 = 1^1 + 6^2 + 7^3 + 6^4 = 1 + 36 + 343 + 1296 = 1676

2427 = 2^1 + 4^2 + 2^3 + 7^4 = 2 + 16 + 8 + 2401 = 2427

The pattern doesn’t apply to any five-digit number in base-10 and six-digit numbers supply only this near miss:

263248 + 1 = 2^1 + 6^2 + 3^3 + 2^4 + 4^5 + 8^6 = 2 + 36 + 27 + 16 + 1024 + 262144 = 263249

But the pattern re-appears among seven-digit numbers:

2646798 = 2^1 + 6^2 + 4^3 + 6^4 + 7^5 + 9^6 + 8^7 = 2 + 36 + 64 + 1296 + 16807 + 531441 + 2097152 = 2646798

Now try some base behaviour. Some power-sums in base-10 are power-sums in another base:

175 = 1^1 + 7^2 + 5^3 = 1 + 49 + 125 = 175

175 = 6D_{[b=27]} = 6^1 + 13^2 = 6 + 169 = 175

1306 = 1^1 + 3^2 + 0^3 + 6^4 = 1 + 9 + 0 + 1296 = 1306

1306 = A[36]_{[b=127]} = 10^1 + 36^2 = 10 + 1296 = 1306

Here is an incomplete list of double-base power-sums:

83 = 1103_{[b=4]} = 1^1 + 1^2 + 0^3 + 3^4 = 1 + 1 + 0 + 81 = 83

83 = 29_{[b=37]} = 2^1 + 9^2 = 2 + 81 = 83

126 = 105_{[b=11]} = 1^1 + 0^2 + 5^3 = 1 + 0 + 125 = 126

126 = 5B_{[b=23]} = 5^1 + 11^2 = 5 + 121 = 126

175 = 1^1 + 7^2 + 5^3 = 1 + 49 + 125 = 175

175 = 6D_{[b=27]} = 6^1 + 13^2 = 6 + 169 = 175

259 = 2014_{[b=5]} = 2^1 + 0^2 + 1^3 + 4^4 = 2 + 0 + 1 + 256 = 259

259 = 3G_{[b=81]} = 3^1 + 16^2 = 3 + 256 = 259

266 = 176_{[b=13]} = 1^1 + 7^2 + 6^3 = 1 + 49 + 216 = 266

266 = AG_{[b=25]} = 10^1 + 16^2 = 10 + 256 = 266

578 = 288_{[b=15]} = 2^1 + 8^2 + 8^3 = 2 + 64 + 512 = 578

578 = 2[24]_{[b=277]} = 2^1 + 24^2 = 2 + 576 = 578

580 = 488_{[b=11]} = 4^1 + 8^2 + 8^3 = 4 + 64 + 512 = 580

580 = 4[24]_{[b=139]} = 4^1 + 24^2 = 4 + 576 = 580

731 = 209_{[b=19]} = 2^1 + 0^2 + 9^3 = 2 + 0 + 729 = 731

731 = 2[27]_{[b=352]} = 2^1 + 27^2 = 2 + 729 = 731

735 = 609_{[b=11]} = 6^1 + 0^2 + 9^3 = 6 + 0 + 729 = 735

735 = 6[27]_{[b=118]} = 6^1 + 27^2 = 6 + 729 = 735

1306 = 1^1 + 3^2 + 0^3 + 6^4 = 1 + 9 + 0 + 1296 = 1306

1306 = A[36]_{[b=127]} = 10^1 + 36^2 = 10 + 1296 = 1306

1852 = 3BC_{[b=23]} = 3^1 + 11^2 + 12^3 = 3 + 121 + 1728 = 1852

1852 = 3[43]_{[b=603]} = 3^1 + 43^2 = 3 + 1849 = 1852

2943 = 3EE_{[b=29]} = 3^1 + 14^2 + 14^3 = 3 + 196 + 2744 = 2943

2943 = [27][54]_{[b=107]} = 27^1 + 54^2 = 27 + 2916 = 2943

Previously pre-posted (please peruse):

• Narcissarithmetic #1

• Narcissarithmetic #2