A334901 Infinitary practical numbers: numbers m such that every number 1 <= k <= isigma(m) is a sum of distinct infinitary divisors of m, where isigma is A049417.

1, 2, 6, 8, 24, 30, 40, 42, 54, 56, 66, 72, 78, 88, 104, 120, 128, 168, 210, 216, 264, 270, 280, 312, 330, 360, 378, 384, 390, 408, 440, 456, 462, 480, 504, 510, 520, 546, 552, 570, 594, 600, 616, 640, 672, 680, 690, 696, 702, 714, 728, 744, 750, 760, 792, 798

Offset: 1

Author

Amiram Eldar, May 16 2020

Keywords

• nonn

Comments

Includes the powers of 2 of the form 2^(2^k - 1) for k = 0, 1, ... (A058891). The other terms are a subset of infinitary abundant numbers (A129656) and infinitary pseudoperfect numbers (A306983).

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000

Crossrefs

The infinitary version of A005153.

Cf. A049417, A058891, A129656, A286652, A306983, A334898.

Programs

Mathematica

bin[n_] := 2^(-1 + Position[Reverse @ IntegerDigits[n, 2], ?(# == 1 &)] // Flatten); f[p, e_] := p^bin[e]; icomp[n_] := Flatten[f @@@ FactorInteger[n]]; fun[p_, e_] := Module[{b = IntegerDigits[e, 2]}, m = Length[b]; Product[If[b[[j]] > 0, 1 + p^(2^(m - j)), 1], {j, 1, m}]]; isigma[1] = 1; isigma[n_] := Times @@ fun @@@ FactorInteger[n]; infPracQ[n_] := Module[{f, p, e, prod = 1, ok = True}, If[n < 1 || (n > 1 && OddQ[n]), False, If[n == 1, True, r = Sort[icomp[n]]; Do[If[r[[i]] > 1 + isigma[prod], ok = False; Break[]]; prod = prod*r[[i]], {i, Length[r]}]; ok]]]; Select[Range[1000], infPracQ]