A007592 -id:A007592 - cp's OEIS frontend

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

Jud McCranie

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

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

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

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

Cf. A034898, A007592, A019279.

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 *)

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

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

More complete name from Jud McCranie, Aug 06 2019

A038536 Odd values of k > 1 for which there are k-hyperperfect numbers.

Original entry on oeis.org

3, 11, 19, 31, 35, 59, 75, 91, 111, 115, 131, 151, 179, 235, 255, 311, 335, 339, 371, 375, 399, 411, 431, 439, 495, 515, 531, 539, 551, 591, 619, 675, 739, 791, 795, 811, 839, 851, 871, 915, 951, 999, 1015, 1035, 1039, 1055, 1071, 1075, 1155, 1231, 1351

Offset: 1

Jud McCranie

nonn

((3*k+1)/2)^2*(3*k+4) is k-hyperperfect.

Richard K. Guy, Unsolved Problems in Number Theory, 3rd Edition, Springer, 2004, Section B2, p. 79.
Joe Roberts, Lure of the Integers, Math. Assoc. America, 1992, p. 177.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Judson S. McCranie, A study of hyperperfect numbers, J. Int. Seq., Vol. 3 (2000), Article 00.1.3.
Daniel Minoli, Issues In Non-Linear Hyperperfect Numbers, Mathematics of Computation, Vol. 34, No. 150, April 1980, pp. 639-645.

Cf. A007592.

q[k_] := Module[{m = (3*k + 1)^2*(3*k + 4)/4}, Divisible[m - 1, DivisorSigma[1, m] - m - 1]]; Select[Range[3, 1500, 2], q] (* Amiram Eldar, May 25 2025 *)

isok(k) = if(k == 1 || !(k % 2), 0, my(m = (3*k + 1)^2*(3*k + 4)/4); !((m-1) % (sigma(m)-m-1))); \\ Amiram Eldar, May 25 2025

A100713 Hyperperfect brilliant numbers.

Original entry on oeis.org

21, 697, 1333, 1909, 3901, 96361, 130153, 163201, 2708413, 2768581, 4013833, 4312681, 4658449, 6392257, 7478041, 8766061, 8883841, 9427657, 9699181, 12064333, 14489437, 15042553, 16260901, 16904101, 18116737, 21396313, 28005301, 29751229, 31837801, 36640993

Offset: 1

Jonathan Vos Post, Dec 11 2004

			21 = 3 * 7, 697 = 17 * 41, 1333 = 31 * 43, 1909 = 23 * 83, 3901 = 47 * 83, 96361 = 173 * 557, 130153 = 157 * 829, 163201 = 293 * 557.
a(2) = 697 because 697 is a 12-hyperperfect number, A028500(2) and is a brilliant number because 697 = 17 * 41.

Richard K. Guy, "Almost Perfect, Quasi-Perfect, Pseudoperfect, Harmonic, Weird, Multiperfect and Hyperperfect Numbers", Section B2 in Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, pp. 45-53, 1994.
Joe Roberts, The Lure of the Integers, Washington, DC: Math. Assoc. Amer., p. 177, 1992.

Amiram Eldar, Table of n, a(n) for n = 1..2678
Judson S. McCranie, A Study of Hyperperfect Numbers. J. Integer Sequences 3, No. 00.1.3, 2000.
Daniel Minoli, Issues in Nonlinear Hyperperfect Numbers, Math. Comput., Vol. 34, No. 150 (1980), pp. 639-645.
Eric Weisstein's World of Mathematics, Hyperperfect Number.

Cf. A007592, A078972, A001358.

a(n) is an element in the intersection of A007592 and A078972. a(n)=m(sigma(a(n))-a(n)-1)+1 for some m>1 and a(n) is a semiprime with the same number of digits in each prime factor.

More terms from Amiram Eldar, Dec 01 2020

A133447 Nonsemiprime hyperperfect numbers.

Original entry on oeis.org

325, 2133, 10693, 16513, 19521, 51301, 159841, 176661, 214273, 306181, 1433701, 1570153, 1950625, 2469601, 2924101, 5199013, 9398593, 10445221, 15407173, 23548753, 28600321, 39147301, 60110701, 62722153, 88347781, 112803841

Offset: 1

Jonathan Vos Post, Dec 22 2007

nonn

The other 25 of the first 35 values of A007592 are all semiprimes A001358.

This sequence excludes perfect numbers (A000396), which are 1-hyperperfect numbers. - Jud McCranie, Mar 23 2025.

			a(1) = 325 = 5^2 * 13.
a(2) = 2133 = 3^3 * 79.
a(3) = 10693 = 17^2 * 37.
a(4) = 16513 = 7^2 * 337.
a(5) = 19521 = 3^4 * 341.
a(6) = 51301 = 29^2 * 61.
a(7) = 159841 = 11^2 * 1321.
a(8) = 176661 = 3^5 * 727.
a(9) = 214273 = 47^2 * 97.
a(10) = 306181 = 53^2 * 109.
a(12) = 1570153 = 13 * 269 * 449. - _Jud McCranie_, Mar 23 2025

Jud McCranie, Table of n, a(n) for n = 1..1167 (terms < 10^15). Earlier terms from Jonathan Vos Post and R. J. Mathar, and a b-file with 192 terms (to 10^12) by Donovan Johnson.

Cf. A001358, A007592, A100959.

a034897 := [] : fd := fopen("b034897.txt",READ) : bf := fscanf(fd,"%d %d") : while nops(bf) <> 0 do a034897 := [op(a034897), op(2,bf) ] ; bf := fscanf(fd,"%d %d") ; od: a007592 := [] : for n in a034897 do m := (n-1)/( numtheory[sigma](n)-n-1) ; if m > 1 then a007592 := [op(a007592),n] ; fi ; od: isA100959 := proc(n) if numtheory[bigomega](n) <> 2 then true ; else false ; end: end: for n in a007592 do if isA100959(n) then printf("%d, ",n) ; fi ; od: # R. J. Mathar, Jan 08 2008

A100959 INTERSECTION A007592.

More terms from R. J. Mathar, Jan 08 2008

A174226 Partial sums of A034897.

Original entry on oeis.org

6, 27, 55, 356, 681, 1177, 1874, 3207, 5116, 7157, 9290, 13191, 21319, 32012, 48525, 68046, 92647, 119624, 170925, 267286, 397439, 557280, 720481, 897142, 1111415, 1361736, 1637569, 1933910, 2240091, 2629684, 3116561, 3612090, 4154503

Offset: 1

Jonathan Vos Post, Mar 12 2010

nonn

Partial sums of hyperperfect numbers. The subsequence of prime values in this partial sum begins: 21319, 92647, 720481.

			a(x) = 6 + 21 + 28 + 301 + 325 + 496 + 697 + 1333 + 1909 + 2041 + 2133 + 3901 + 8128 + 10693 + 16513 + 19521 + 24601 = 92647 is prime.

Cf. A034897, A034898, A007592, A019279.

a(n) = SUM[i=1..n] A034897(i) = SUM[i=1..n] {m such that k*sigma(m) = (k+1)*m + k - 1 for some positive integer k, and sigma being the divisor function A000005; noting that k=1 gives perfect numbers}.

cp's OEIS Frontend

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

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A038536 Odd values of k > 1 for which there are k-hyperperfect numbers.

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A100713 Hyperperfect brilliant numbers.

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A133447 Nonsemiprime hyperperfect numbers.

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A174226 Partial sums of A034897.

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