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).

24, 40, 56, 80, 88, 104, 120, 224, 360, 432, 672, 832, 992, 1008, 1296, 1456, 1504, 1584, 1888, 1952, 2016, 2160, 2800, 3800, 5624, 5800, 7424, 7616, 9112, 10080, 11096, 13736, 15872, 16256, 17816, 22848, 24448, 28544, 30592, 32128, 33728, 51136, 62464, 66368

Offset

1,1

Comments

The bi-unitary version of A181595.

Links

Amiram Eldar, Table of n, a(n) for n = 1..183

Example

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.

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

Prog

(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, sig - 2*n); \\ Michel Marcus, Apr 27 2018

Crossrefs

Cf. A153501, A181595, A188999, A292982, A303358.

Sequence in context: A065127 A065036 A329880 * A340747 A340746 A043119

Adjacent sequences: A303356 A303357 A303358 * A303360 A303361 A303362

Keyword

nonn

Author

Amiram Eldar and Michael De Vlieger, Apr 22 2018

Status

approved