A034897 Hyperperfect numbers: x such that x = 1 + k*(sigma(x)-x-1) for some k > 0.

6, 21, 28, 301, 325, 496, 697, 1333, 1909, 2041, 2133, 3901, 8128, 10693, 16513, 19521, 24601, 26977, 51301, 96361, 130153, 159841, 163201, 176661, 214273, 250321, 275833, 296341, 306181, 389593, 486877, 495529, 542413, 808861, 1005421, 1005649, 1055833

Offset: 1

Author

Jud McCranie

Keywords

• nonn
• nice

Comments

k=1 gives the perfect numbers, A000396. For a general k, they are called k-hyperperfect. - Jud McCranie, Aug 06 2019

There are 105200 hyperperfect numbers < 10^15. a(105200)=999990080853493. - Jud McCranie, Mar 22 2025

Examples

			21 = 1 + 2*(sigma(21)-21-1), so 21 is 2-hyperperfect. - _Jud McCranie_, Aug 06 2019
		

References

• R. K. Guy, Unsolved Problems in Number Theory, Sect. B2.
• J. Roberts, Lure of the Integers, see Integer 28, p. 177.

Links

• Jud McCranie and Donovan Johnson, Table of n, a(n) for n = 1..10000 (first 2190 terms from Jud McCranie)
• J. S. McCranie, A study of hyperperfect numbers, J. Int. Seqs. Vol. 3 (2000) #P00.1.3.
• Eric Weisstein's World of Mathematics, Hyperperfect Number.
• Wikipedia, Hyperperfect number

Crossrefs

Cf. A034898, A007592, A019279.

Programs

Mathematica

hpnQ[n_]:=Module[{c=DivisorSigma[1,n]-n-1},c>0&&IntegerQ[(n-1)/c]]; Select[Range[2,809000],hpnQ] (* Harvey P. Dale, Jan 17 2012 *)

PARI

forcomposite(n=2, 2*10^6, if(1==Mod(n, sigma(n)-n-1), print1(n", "))) \\ Hans Loeblich, May 07 2019

Python

from itertools import count, islice
from sympy import isprime, divisor_sigma
def A034897_gen(): # generator of terms
    return (n for n in count(2) if not isprime(n) and (n-1) % (divisor_sigma(n)-n-1) == 0)
A034897_list = list(islice(A034897_gen(),10)) # Chai Wah Wu, Feb 18 2022

Extensions

More complete name from Jud McCranie, Aug 06 2019