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%I #42 Mar 01 2024 02:05:34
%S 12,30,56,306,380,858,992,1722,2552,2862,16256,30704,66198,73712,
%T 86142,249500,629802,1703872,6127552,16191736,19127502,35359900,
%U 67100672,101999900,172173762,182552538,266677578,575688042,1180712682,2214408306,6179139056,17179738112,21083999500
%N Prime power Giuga numbers: composite numbers n > 1 such that -1/n + sum 1/p^k = 1, where the sum is over the prime powers p^k dividing n.
%C Since Giuga numbers (A007850) must be squarefree, it follows all Giuga numbers are contained in this sequence.
%C The number 2^k (2^k-1) is in this sequence whenever 2^k-1 is a Mersenne prime (A000668).
%H John Machacek, <a href="https://arxiv.org/abs/1706.01008">Egyptian Fractions and Prime Power Divisors</a>, arXiv:1706.01008 [math.NT], 2017.
%e n = 12 is in the sequence because -1/12 + 1/2 + 1/2^2 + 1/3 = 1.
%e n = 18 is NOT in the sequence because -1/18 + 1/2 + 1/3 + 1/3^2 != 1.
%t ok[n_] := Total[n/Flatten@ Table[e[[1]] ^ Range[e[[2]]], {e, FactorInteger@ n}]] - 1 == n; Select[Range[10^5], ok] (* _Giovanni Resta_, May 27 2017 *)
%Y Cf. A000668, A007850.
%K nonn
%O 1,1
%A _John Machacek_, May 27 2017
%E a(20)-a(31) from _Giovanni Resta_, May 27 2017
%E a(32)-a(33) from _Giovanni Resta_, Jun 26 2017
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