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A334974 Infinitary admirable numbers: numbers k such that there is a proper infinitary divisor d of k such that isigma(k) - 2*d = 2*k, where isigma is the sum of infinitary divisors function (A049417).

%I A334974 #8 May 23 2020 05:44:07
%S A334974 24,30,40,42,54,56,66,70,78,88,96,102,104,114,120,138,150,174,186,222,
%T A334974 246,258,270,282,294,318,354,360,366,402,420,426,438,474,486,498,534,
%U A334974 540,582,606,618,630,642,654,660,678,726,762,780,786,822,834,894,906,942
%N A334974 Infinitary admirable numbers: numbers k such that there is a proper infinitary divisor d of k such that isigma(k) - 2*d = 2*k, where isigma is the sum of infinitary divisors function (A049417).
%C A334974 Equivalently, numbers that are equal to the sum of their proper infinitary divisors, with one of them taken with a minus sign.
%C A334974 Admirable numbers (A111592) whose number of divisors is a power of 2 (A036537) are also infinitary admirable numbers, since all of their divisors are infinitary. Terms with number of divisors that is not a power of 2 are 96, 150, 294, 360, 420, 486, 540, 630, 660, 726, 780, 960, 990, ...
%H A334974 Amiram Eldar, <a href="/A334974/b334974.txt">Table of n, a(n) for n = 1..10000</a>
%e A334974 150 is in the sequence since 150 = 1 + 2 + 3 - 6 + 25 + 50 + 75 is the sum of its proper infinitary divisors with one of them, 6, taken with a minus sign.
%t A334974 fun[p_, e_] := Module[{b = IntegerDigits[e, 2], m}, 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]; infDivQ[n_, 1] = True; infDivQ[n_, d_] := BitAnd[IntegerExponent[n, First /@ (f = FactorInteger[d])], (e = Last /@ f)] == e; infAdmQ[n_] := (ab = isigma[n] - 2 n) > 0 && EvenQ[ab] && ab/2 < n && Divisible[n, ab/2] && infDivQ[n, ab/2]; Select[Range[1000], infAdmQ]
%Y A334974 The infinitary version of A111592.
%Y A334974 Subsequence of A129656.
%Y A334974 Cf. A036537, A049417, A077609, A328328, A334972.
%K A334974 nonn
%O A334974 1,1
%A A334974 _Amiram Eldar_, May 18 2020