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

1,1

Comments

k=1 gives the perfect numbers, A000396. For a general k, they are called k-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

Example

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

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 *)

Prog

(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

Crossrefs

Cf. A034898, A007592, A019279.

Sequence in context: A132184 A143322 A246544 * A347875 A287165 A280296

Adjacent sequences: A034894 A034895 A034896 * A034898 A034899 A034900

Keyword

nonn,nice

Author

Jud McCranie

Extensions

More complete name from Jud McCranie, Aug 06 2019

Status

approved