%I #10 Mar 31 2023 06:56:39
%S 1,2,3,6,8,12,15,24,30,40,42,56,60,72,84,105,108,120,132,135,156,165,
%T 168,195,210,216,240,255,264,270,280,312,330,360,378,384,390,408,420,
%U 440,456,462,480,504,510,520,540,546,552,570,600,616,640,660,672,680,690
%N Infinitary phi-practical numbers: numbers m such that each k <= m is a subsum of a the multiset {iphi(d) : d infinitary divisor of m}, where iphi is an infinitary analog of Euler's phi function (A091732).
%H Amiram Eldar, <a href="/A361922/b361922.txt">Table of n, a(n) for n = 1..10000</a>
%t f[p_, e_] := p^(2^(-1 + Position[Reverse@IntegerDigits[e, 2], 1]));
%t iphi[1] = 1; iphi[n_] := Times @@ (Flatten@ (f @@@ FactorInteger[n]) - 1);
%t idivs[n_] := Sort@ Flatten@ Outer[Times, Sequence @@ (FactorInteger[n] /. {p_, m_Integer} :> p^Select[Range[0, m], BitOr[m, #] == m &])]; idivs[1] = {1};
%t iPhiPracticalQ[n_] := Module[{s = Sort@ Map[iphi, idivs[n]], ans = True}, Do[If[s[[j]] > Sum[s[[i]], {i, 1, j - 1}] + 1, ans = False; Break[]], {j, 1, Length[s]}]; ans]; Select[Range[700], iPhiPracticalQ]
%Y Cf. A077609, A091732.
%Y Similar sequences: A260653, A286906, A334901.
%K nonn
%O 1,2
%A _Amiram Eldar_, Mar 30 2023