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A334972 Bi-unitary admirable numbers: numbers k such that there is a proper bi-unitary divisor d of k such that bsigma(k) - 2*d = 2*k, where bsigma is the sum of bi-unitary divisors function (A188999).

%I A334972 #10 May 23 2020 05:44:11
%S A334972 24,30,40,42,48,54,56,66,70,78,80,88,102,104,114,120,138,150,162,174,
%T A334972 186,222,224,246,258,270,282,294,318,354,360,366,402,420,426,438,448,
%U A334972 474,498,534,540,582,606,618,630,642,654,660,672,678,720,726,762,780,786
%N A334972 Bi-unitary admirable numbers: numbers k such that there is a proper bi-unitary divisor d of k such that bsigma(k) - 2*d = 2*k, where bsigma is the sum of bi-unitary divisors function (A188999).
%C A334972 Equivalently, numbers that are equal to the sum of their proper bi-unitary divisors, with one of them taken with a minus sign.
%C A334972 Admirable numbers (A111592) that are exponentially odd (A268335) are also bi-unitary admirable numbers since all of their divisors are bi-unitary. Terms that are not exponentially odd are 48, 80, 150, 162, 294, 360, 420, 448, 540, 630, 660, 720, 726, 780, 832, 990, ...
%H A334972 Amiram Eldar, <a href="/A334972/b334972.txt">Table of n, a(n) for n = 1..10000</a>
%e A334972 48 is in the sequence since 48 = 1 + 2 + 3 - 6 + 8 + 16 + 24 is the sum of its proper bi-unitary divisors with one of them, 6, taken with a minus sign.
%t A334972 fun[p_, e_] := If[OddQ[e], (p^(e + 1) - 1)/(p - 1), (p^(e + 1) - 1)/(p - 1) - p^(e/2)]; bsigma[1] = 1; bsigma[n_] := Times @@ (fun @@@ FactorInteger[n]); buDivQ[n_, 1] = True; buDivQ[n_, div_] := If[Mod[#2, #1] == 0, Last@Apply[Intersection, Map[Select[Divisors[#], Function[d, CoprimeQ[d, #/d]]] &, {#1, #2/#1}]] == 1, False] & @@ {div, n}; buAdmQ[n_] := (ab = bsigma[n] - 2 n) > 0 && EvenQ[ab] && ab/2 < n && Divisible[n, ab/2] && buDivQ[n, ab/2]; Select[Range[1000], buAdmQ]
%Y A334972 The bi-unitary version of A111592.
%Y A334972 Subsequence of A292982.
%Y A334972 Cf. A188999, A222266, A268335, A328328, A334974.
%K A334972 nonn
%O A334972 1,1
%A A334972 _Amiram Eldar_, May 18 2020