A306987 Primitive abundant numbers (A071395) that are pseudoperfect (A005835).

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

Author

Amiram Eldar, Mar 18 2019

Keywords

• nonn

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

• Amiram Eldar, Table of n, a(n) for n = 1..10000
• S. J. Benkoski and P. Erdős, On weird and pseudoperfect numbers, Math. Comp. 28 (1974), 617-623. Corrigendum: Math. Comp. 29 (1975), 673-674.

Crossrefs

Cf. A005835, A006036, A071395.

Programs

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 *)