A034897 - OEIS
A034897
Hyperperfect numbers: x such that x = 1 + k*(sigma(x)-x-1) for some k > 0.
10
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
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
REFERENCES
R. K. Guy, Unsolved Problems in Number Theory, Sect. B2.
J. Roberts, Lure of the Integers, see Integer 28, p. 177.
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)
EXTENSIONS
More complete name from Jud McCranie, Aug 06 2019