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A306987
Primitive abundant numbers (A071395) that are pseudoperfect (A005835).
2
20, 88, 104, 272, 304, 368, 464, 550, 572, 650, 748, 945, 1184, 1312, 1376, 1430, 1504, 1575, 1696, 1870, 1888, 1952, 2002, 2090, 2205, 2210, 2470, 2530, 2584, 2990, 3128, 3190, 3230, 3410, 3465, 3496, 3770, 3944, 4070, 4095, 4216, 4288, 4408, 4510, 4544, 4672
OFFSET
1,1
COMMENTS
By definition these numbers are also primitive pseudoperfect (A006036).
Benkoski and Erdős proved that this sequence is infinite, since it includes all the numbers of the form 2^k * p with p a prime such that 2^k < p < 2^(k+1).
LINKS
S. J. Benkoski and P. Erdős, On weird and pseudoperfect numbers, Math. Comp. 28 (1974), 617-623. Corrigendum: Math. Comp. 29 (1975), 673-674.
MATHEMATICA
paQ[n_]:=DivisorSigma[1, n] > 2n && Times @@ Boole@ Map[DivisorSigma[1, #] < 2 # &, Most@ Divisors@ n] == 1; psQ[n_]:=Module[{d= Most[Divisors[n] ]}, SeriesCoefficient[Series[Product[1 + x^d[[i]], {i, Length[d]}], {x, 0, n}], n] > 0]; Select[Range[5000], paQ[#]&&psQ[#]&] (* after Michael De Vlieger at A071395 and T. D. Noe at A005835 *)
CROSSREFS
KEYWORD
nonn
AUTHOR
Amiram Eldar, Mar 18 2019
STATUS
approved