A225791
Numbers n such that the sum of first n prime powers (A025475) is divisible by n.
Original entry on oeis.org
1, 9, 54, 85, 853, 1162, 4209, 11566, 20060, 68048, 76221, 441294, 3007789, 4521955, 39443840, 39755851
Offset: 1
The sum of first 9 prime powers is 1 + 4 + 8 + 9 + 16 + 25 + 27 + 32 + 49 = 171. Because 171 is divisible by 9, the latter is in the sequence.
A227032
Numbers k such that the sum of the first k prime powers > 1 is divisible by k.
Original entry on oeis.org
1, 3, 36, 75, 96, 692, 732, 798, 2407, 3102, 3941, 4003, 101423, 131281, 337708, 399418, 460530, 480328, 594577, 26121722, 287922744, 702590009, 870360308, 18056232172, 35392830431, 73730356109, 169558413390, 268671255173
Offset: 1
The first 3 prime powers > 1 are 2, 3, 2^2 and 2+3+2^2 is divisible by 3, so 3 is a term.
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seq = {}; s = n = 0; p = 1; While[n < 10^4, If[Length@FactorInteger[++p] == 1 && Mod[s += p, ++n] == 0, AppendTo[seq, n]]]; seq
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