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Hyperperfect numbers: k = m*(sigma(k) - k - 1) + 1 for some m > 1.
(Formerly M5113)
6

%I M5113 #43 Dec 16 2021 22:32:00

%S 21,301,325,697,1333,1909,2041,2133,3901,10693,16513,19521,24601,

%T 26977,51301,96361,130153,159841,163201,176661,214273,250321,275833,

%U 296341,306181,389593,486877,495529,542413,808861,1005421,1005649,1055833,1063141,1232053

%N Hyperperfect numbers: k = m*(sigma(k) - k - 1) + 1 for some m > 1.

%D D. Minoli, Sufficient Forms For Generalized Perfect Numbers, Ann. Fac. Sciences, Univ. Nation. Zaire, Section Mathem; Vol. 4, No. 2, Dec 1978, pp. 277-302.

%D D. Minoli, New Results For Hyperperfect Numbers, Abstracts American Math. Soc., October 1980, Issue 6, Vol. 1, pp. 561.

%D J. Roberts, Lure of the Integers, Math. Assoc. America, 1992, p. 177.

%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

%H Donovan Johnson, <a href="/A007592/b007592.txt">Table of n, a(n) for n = 1..10000</a>

%H Mariano Garcia, <a href="http://www.fq.math.ca/Papers1/42-4/quartgarcia04_2004.pdf">Hyperperfect Numbers with Five and Six Different Prime Factors</a>, Fib. Quart. 42, no. 4, Nov. 2004, pp. 292-294.

%H J. S. McCranie, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL3/mccranie.html">A study of hyperperfect numbers</a>, J. Int. Seqs. Vol. 3 (2000) #P00.1.3.

%H D. Minoli and Robert Bear, <a href="http://www.pme-math.org/journal/issues/PMEJ.Vol.6.No.3.pdf">Hyperperfect Numbers</a>, Pi Mu Epsilon Journal, Fall 1975, pp. 153-157.

%H Daniel Minoli, W. Nakamine, <a href="http://dx.doi.org/10.1109/ICASSP.1980.1170906">Mersenne Numbers Rooted On 3 For Number Theoretic Transforms</a>, 1980 IEEE International Conf. on Acoust., Speech and Signal Processing.

%H Daniel Minoli, <a href="http://dx.doi.org/10.1090/S0025-5718-1980-0559206-9">Issues In Non-Linear Hyperperfect Numbers</a>, Mathematics of Computation, Vol. 34, No. 150, April 1980, pp. 639-645.

%H D. Minoli, <a href="http://www.fq.math.ca/Scanned/19-1/minoli.pdf">Structural Issues For Hyperperfect Numbers</a>, Fibonacci Quarterly, Feb. 1981, Vol. 19, No. 1, pp. 6-14.

%H Herman J. J. te Riele, <a href="http://dx.doi.org/10.1090/S0025-5718-1981-0595066-9">Hyperperfect numbers with three different prime factors</a>, Math. Comp. 36 (1981), 297-298.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/HyperperfectNumber.html">Hyperperfect Number.</a>

%t hpnQ[n_]:=Module[{den=DivisorSigma[1,n]-n-1,c},If[den!=0,c=(n-1)/den, c=Pi];IntegerQ[c]&&c>1]; Select[Range[1250000],hpnQ] (* _Harvey P. Dale_, Aug 11 2012 *)

%Y See A034897 (for m >= 1).

%K nonn,nice

%O 1,1

%A _N. J. A. Sloane_

%E More terms from _Jud McCranie_, Oct 15 1997