OFFSET
1,1
COMMENTS
First differs from 8 * A381313 at n = 32.
Numbers k such that the sum of the divisors of k that are both coreful and bi-unitary is larger than 2*k.
All the terms are both bi-unitary abundant numbers (A292982) and coreful abundant numbers (A308053) but this sequence is not the intersection of these two sequences, which includes the terms 72, 288, 360, 504, 540, ... that are not in this sequence.
If k and m are coprime terms, then k*m is also a term.
In particular, if k is a term, and m is a squarefree number coprime to k, then k*m is also a term. The primitive terms in this sequence (A391507) are the powerful (A001694) terms.
The asymptotic density of this sequence is Sum_{n>=1} f(A391507(n)) = 0.00586028..., where f(n) = (6/(Pi^2*n)) * Product_{prime p|n} (p/(p+1)).
The least odd term is a(6784) = 1157625 = (3 * 5 * 7)^3.
The least term that is coprime to 6 is a(3.01...*10^20) = 51404758182902197698625 = (5 * 7 * 11 * 13 * 17 * 19 * 23)^3.
LINKS
Amiram Eldar, Table of n, a(n) for n = 1..10000
MATHEMATICA
f[p_, e_] := (p^(e+1) - 1)/(p - 1) - 1 - If[OddQ[e], 0, p^(e/2)]; q[1] = False; q[n_] := Times @@ f @@@ FactorInteger[n] > 2*n; Select[Range[8000], q]
PROG
(PARI) isok(k) = {my(f = factor(k)); prod(i = 1, #f~, (f[i, 1]^(f[i, 2] + 1) - 1)/(f[i, 1] - 1) - 1 - if(f[i, 2] % 2, 0, f[i, 1]^(f[i, 2]/2))) > 2*k; }
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Amiram Eldar, Dec 11 2025
STATUS
approved