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).
1, 2, 8, 10, 12, 32, 112, 128, 136, 144, 152, 184, 512, 1088, 2048, 2144, 2272, 2528, 2736, 3248, 3312, 4592, 7936, 8192, 9800, 11800, 17176, 18632, 18904, 22984, 32768, 32896, 33664, 34688, 49024, 57152, 77248, 85952, 131072, 176400, 212400, 309168, 335376
Offset: 1
Keywords
Examples
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.
Links
- Amiram Eldar, Table of n, a(n) for n = 1..171
Programs
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Mathematica
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 -
PARI
udivs(n) = {my(d = divisors(n)); select(x->(gcd(x, n/x)==1), d); } gcud(n, m) = vecmax(setintersect(udivs(n), udivs(m))); biudivs(n) = select(x->(gcud(x, n/x)==1), divisors(n)); isok(n) = my(divs = biudivs(n), sig = vecsum(divs)); (sig < 2*n) && vecsearch(divs, 2*n-sig); \\ Michel Marcus, Apr 27 2018
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