A309568 Bi-unitary k-hyperperfect numbers: numbers m such that m = 1 + k * (bsigma(m) - m - 1) where bsigma(m) is the sum of bi-unitary divisors of m (A188999) and k >= 1 is an integer.
6, 21, 52, 60, 90, 301, 657, 697, 1333, 1909, 2041, 2133, 3901, 15025, 24601, 26977, 96361, 130153, 163201, 176661, 250321, 275833, 296341, 389593, 486877, 495529, 542413, 808861, 1005421, 1005649, 1055833, 1063141, 1232053, 1246417, 1284121, 1357741, 1403221
Offset: 1
Keywords
Examples
21 is in the sequence since bsigma(21) = 32 and 21 = 1 + 2 * (32 - 21 - 1).
Links
- Amiram Eldar, Table of n, a(n) for n = 1..1000
- József Sándor and Mihály Bencze, On modified hyperperfect numbers, Research report collection, Vol. 8, No. 2 (2005).
Programs
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Mathematica
fun[p_, e_] := If[OddQ[e], (p^(e+1)-1)/(p-1), (p^(e+1)-1)/(p-1)-p^(e/2)]; bsigma[1] = 1; bsigma[n_] := Times @@ (fun @@@ FactorInteger[n]); hpnQ[n_] := (c = bsigma[n]-n-1) > 0 && Divisible[n-1, c]; Select[Range[10^5], hpnQ]
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