A007592 Hyperperfect numbers: k = m*(sigma(k) - k - 1) + 1 for some m > 1.

21, 301, 325, 697, 1333, 1909, 2041, 2133, 3901, 10693, 16513, 19521, 24601, 26977, 51301, 96361, 130153, 159841, 163201, 176661, 214273, 250321, 275833, 296341, 306181, 389593, 486877, 495529, 542413, 808861, 1005421, 1005649, 1055833, 1063141, 1232053

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• nice

References

• 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. Minoli, New Results For Hyperperfect Numbers, Abstracts American Math. Soc., October 1980, Issue 6, Vol. 1, pp. 561.
• J. Roberts, Lure of the Integers, Math. Assoc. America, 1992, p. 177.
• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

• Donovan Johnson, Table of n, a(n) for n = 1..10000
• Mariano Garcia, Hyperperfect Numbers with Five and Six Different Prime Factors, Fib. Quart. 42, no. 4, Nov. 2004, pp. 292-294.
• J. S. McCranie, A study of hyperperfect numbers, J. Int. Seqs. Vol. 3 (2000) #P00.1.3.
• D. Minoli and Robert Bear, Hyperperfect Numbers, Pi Mu Epsilon Journal, Fall 1975, pp. 153-157.
• Daniel Minoli, W. Nakamine, Mersenne Numbers Rooted On 3 For Number Theoretic Transforms, 1980 IEEE International Conf. on Acoust., Speech and Signal Processing.
• Daniel Minoli, Issues In Non-Linear Hyperperfect Numbers, Mathematics of Computation, Vol. 34, No. 150, April 1980, pp. 639-645.
• D. Minoli, Structural Issues For Hyperperfect Numbers, Fibonacci Quarterly, Feb. 1981, Vol. 19, No. 1, pp. 6-14.
• Herman J. J. te Riele, Hyperperfect numbers with three different prime factors, Math. Comp. 36 (1981), 297-298.
• Eric Weisstein's World of Mathematics, Hyperperfect Number.

Crossrefs

See A034897 (for m >= 1).

Programs

Mathematica

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 *)

Extensions

More terms from Jud McCranie, Oct 15 1997