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

1,1

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, ...

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).

Example

21 is in the sequence since bsigma(21) = 32 and 21 = 1 + 2 * (32 - 21 - 1).

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]

Crossrefs

Cf. A034897, A188999, A225150.

Sequence in context: A341066 A263418 A097124 * A244906 A276072 A135454

Adjacent sequences: A309565 A309566 A309567 * A309569 A309570 A309571

Keyword

nonn

Author

Amiram Eldar, Aug 08 2019

Status

approved