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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.

%I A309568 #15 Aug 19 2019 03:16:58
%S A309568 6,21,52,60,90,301,657,697,1333,1909,2041,2133,3901,15025,24601,26977,
%T A309568 96361,130153,163201,176661,250321,275833,296341,389593,486877,495529,
%U A309568 542413,808861,1005421,1005649,1055833,1063141,1232053,1246417,1284121,1357741,1403221
%N 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.
%C A309568 The bi-unitary version of A034897.
%C A309568 The only bi-unitary 1-hyperperfect numbers are 6, 60, and 90 (the bi-unitary perfect numbers).
%C A309568 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, ...
%H A309568 Amiram Eldar, <a href="/A309568/b309568.txt">Table of n, a(n) for n = 1..1000</a>
%H A309568 József Sándor and Mihály Bencze, <a href="http://rgmia.org/papers/v8n2/mhpn.pdf">On modified hyperperfect numbers</a>, Research report collection, Vol. 8, No. 2 (2005).
%e A309568 21 is in the sequence since bsigma(21) = 32 and 21 = 1 + 2 * (32 - 21 - 1).
%t A309568 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]
%Y A309568 Cf. A034897, A188999, A225150.
%K A309568 nonn
%O A309568 1,1
%A A309568 _Amiram Eldar_, Aug 08 2019