cp's OEIS Frontend

A303359 Bi-unitary near-perfect numbers: bi-unitary abundant numbers k such that the abundance d = bsigma(k) - 2*k is a bi-unitary divisor of k, where bsigma(k) is the sum of bi-unitary divisors of k (A188999).

%I A303359 #15 Jan 25 2020 06:18:29
%S A303359 24,40,56,80,88,104,120,224,360,432,672,832,992,1008,1296,1456,1504,
%T A303359 1584,1888,1952,2016,2160,2800,3800,5624,5800,7424,7616,9112,10080,
%U A303359 11096,13736,15872,16256,17816,22848,24448,28544,30592,32128,33728,51136,62464,66368
%N A303359 Bi-unitary near-perfect numbers: bi-unitary abundant numbers k such that the abundance d = bsigma(k) - 2*k is a bi-unitary divisor of k, where bsigma(k) is the sum of bi-unitary divisors of k (A188999).
%C A303359 The bi-unitary version of A181595.
%H A303359 Amiram Eldar, <a href="/A303359/b303359.txt">Table of n, a(n) for n = 1..183</a>
%e A303359 24 is in the sequence since the sum of its bi-unitary divisors is 1 + 2 + 3 + 4 + 6 + 8 + 12 + 24 = 60 and 60 - 2*24 = 12 is a bi-unitary divisor of 24.
%t A303359 f[n_] := Select[Divisors[n], Function[d, CoprimeQ[d, n/d]]]; bsigma[m_] := DivisorSum[m, # &, Last@Intersection[f@#, f[m/#]] == 1 &]; biunitaryDivisorQ[ div_, n_] := If[Mod[#2, #1]==0, Last@Apply[Intersection, Map[Select[Divisors[#], Function[d, CoprimeQ[d, #/d]]]&, {#1, #2/#1}]]==1, False]& @@{div, n}; aQ[n_] := Module[{d=bsigma[n]-2n},If[d<=0, False,biunitaryDivisorQ[d,n]]]; s={}; Do[If[ aQ[n], AppendTo[s,n] ], {n, 1, 10000}]; s
%o A303359 (PARI) udivs(n) = {my(d = divisors(n)); select(x->(gcd(x, n/x)==1), d); }
%o A303359 gcud(n, m) = vecmax(setintersect(udivs(n), udivs(m)));
%o A303359 biudivs(n) = select(x->(gcud(x, n/x)==1), divisors(n));
%o A303359 isok(n) = my(divs = biudivs(n), sig = vecsum(divs)); (sig > 2*n) && vecsearch(divs, sig - 2*n); \\ _Michel Marcus_, Apr 27 2018
%Y A303359 Cf. A153501, A181595, A188999, A292982, A303358.
%K A303359 nonn
%O A303359 1,1
%A A303359 _Amiram Eldar_ and _Michael De Vlieger_, Apr 22 2018