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A303358 Bi-unitary deficient-perfect numbers: bi-unitary deficient numbers k for such that 2*k - bsigma(k) is a bi-unitary divisor of k, where bsigma(k) is the sum of bi-unitary divisors of k (A188999). 2

%I

%S 1,2,8,10,12,32,112,128,136,144,152,184,512,1088,2048,2144,2272,2528,

%T 2736,3248,3312,4592,7936,8192,9800,11800,17176,18632,18904,22984,

%U 32768,32896,33664,34688,49024,57152,77248,85952,131072,176400,212400,309168,335376

%N Bi-unitary deficient-perfect numbers: bi-unitary deficient numbers k for such that 2*k - bsigma(k) is a bi-unitary divisor of k, where bsigma(k) is the sum of bi-unitary divisors of k (A188999).

%C The bi-unitary version of A271816.

%C Includes all the odd powers of 2 (A004171).

%H Amiram Eldar, <a href="/A303358/b303358.txt">Table of n, a(n) for n = 1..171</a>

%e 112 is in the sequence since the sum of its bi-unitary divisors is 1 + 2 + 7 + 8 + 14 + 16 + 56 + 112 = 216 and 2*112 - 216 = 8 is a bi-unitary divisor of 112.

%t 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=2n-bsigma[n]},If[d<=0, False,biunitaryDivisorQ[d,n]]]; s={}; Do[ If[aQ[n], AppendTo[s,n]], {n, 1, 10000}]; s

%o (PARI) udivs(n) = {my(d = divisors(n)); select(x->(gcd(x, n/x)==1), d); }

%o gcud(n, m) = vecmax(setintersect(udivs(n), udivs(m)));

%o biudivs(n) = select(x->(gcud(x, n/x)==1), divisors(n));

%o isok(n) = my(divs = biudivs(n), sig = vecsum(divs)); (sig < 2*n) && vecsearch(divs, 2*n-sig); \\ _Michel Marcus_, Apr 27 2018

%Y Cf. A004171, A188999, A271816, A292982, A303359.

%K nonn

%O 1,2

%A _Amiram Eldar_ and _Michael De Vlieger_, Apr 22 2018

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Last modified February 26 14:18 EST 2021. Contains 341632 sequences. (Running on oeis4.)