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

Author

Amiram Eldar, Aug 08 2019

Keywords

• nonn

Comments

The bi-unitary version of A034897.

The only bi-unitary 1-hyperperfect numbers are 6, 60, and 90 (the bi-unitary perfect numbers).

The corresponding k values are 1, 2, 3, 1, 1, 6, 8, 12, 18, 18, 12, 2, 30, 24, 60, 48, 132, 132, 192, 2, 168, 108, 66, 252, 78, 132, 342, 366, 390, 168, 348, 282, 498, 552, 540, 30, 546, ...

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).

Crossrefs

Cf. A034897, A188999, A225150.

Programs

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]