A066290 Numbers m such that DivisorSigma(4*k-2, m) mod m = 0 holds presumably for all k; that is, (4k-2)-power-sums of divisors of m are divisible by m for all k.
1, 10, 60, 65, 130, 150, 260, 780, 1105, 2210, 4420, 8840, 13260, 19720, 20737, 32045, 41474, 55250, 64090, 82948, 103685, 128180, 207370, 207553, 221000, 248844, 256360, 295800, 331500, 352529, 384540, 414740, 415106, 450840, 512720, 705058, 829480, 830212
Offset: 1
Keywords
Examples
Tested for each m and k < 200. Proof for several values of k seems not so tedious because the number of divisors of the terms of the sequence is not so large: {1, 4, 12, 4, 8, 12, 12, 24, 8, 16, 24, 32, 48, 32, 4, 16, 8, 32, 32, 12, 8, 48, 16, 8, 64, 24, 64, 96, 96, 8, 96, 24, 16, 96, 80, 16, 32, 24}.
Programs
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Mathematica
lastSeq = {}; max = 100; While[seq = Reap[For[n = 1, n < 10^6, n++, If[AllTrue[Range[max], Mod[DivisorSigma[4 # - 2, n], n] == 0&], Print[n]; Sow[n]]]][[2, 1]]; seq != lastSeq, lastSeq = seq; max = max + 100; Print["max = ", max]]; seq (* Jean-François Alcover, Oct 02 2016 *)
Formula
DivisorSigma(4k-2, m)/m is an integer for k = 1, 2, 3, .., 200, ...
Extensions
More terms from Jean-François Alcover, Oct 02 2016