A302994 Number of bi-unitary abundant numbers < 10^n.

0, 14, 147, 1553, 15450, 155395, 1549818, 15498814, 155079196, 1550331185, 15503061466, 155037242668, 1550370696100, 15503650949671, 155036854371220, 1550366484701654, 15503648102080675

Offset: 1

Author

Amiram Eldar, Apr 17 2018

Keywords

• nonn
• more

Crossrefs

Cf. A188999, A292982, A302992.

Programs

Mathematica

f[n_] := Select[Divisors[n], Function[d, CoprimeQ[d, n/d]]]; bsigma[m_] :=
DivisorSum[m, # &, Last@Intersection[f@#, f[m/#]] == 1 &]; babQ[n_] := bsigma[n] > 2 n; c = 0; k = 1; seq={}; Do[While[k < 10^n, If[babQ[k], c++]; k++]; AppendTo[seq, c], {n, 1, 5}]; seq

PARI

biusigma(n) = {f = factor(n); for (i=1, #f~, p = f[i, 1]; e = f[i, 2]; f[i, 1] = if (e % 2, (p^(e+1)-1)/(p-1), (p^(e+1)-1)/(p-1) -p^(e/2)); f[i, 2] = 1; ); factorback(f); }
a(n) = sum(k=1, 10^n-1, biusigma(k) > 2*k); \\ Michel Marcus, Apr 17 2018

Formula

Conjecture: Lim_{n->oo} a(n)/10^n = 0.15... is the density of bi-unitary abundant numbers.

Extensions

a(8)-a(17) from Hiroaki Yamanouchi, Aug 24 2018