A019279 -id:A019279 - cp's OEIS frontend

A019285 Let sigma_m (n) be result of applying sum-of-divisors function m times to n; call n (m,k)-perfect if sigma_m (n) = k*n; sequence gives the (2,8)-perfect numbers.

60, 240, 960, 4092, 16368, 58254, 61440, 65472, 116508, 466032, 710400, 983040, 1864128, 3932160, 4190208, 67043328, 119304192, 268173312, 1908867072, 7635468288, 16106127360, 711488165526, 1098437885952, 1422976331052

Offset: 1

N. J. A. Sloane

If 2^p-1 is a Mersenne prime greater than 3 then m = 15*2^(p-1) is in the sequence. Because sigma(sigma(m)) = sigma(15*2^(p-1)) = sigma(24*(2^p-1)) = 60*2^p = 8*(15*2^(p-1)) = 8*m. So for n>1 15/2*(A000668(n)+1) is in the sequence. 60, 240, 960, 61440, 983040, 3932160, 16106127360 and 1729382256910270464042 are such terms. - Farideh Firoozbakht, Dec 05 2005
See also the Cohen-te Riele links under A019276.
No other terms < 5*10^11. - Jud McCranie, Feb 08 2012
1422976331052 is also a term. See comment in A019278. - Michel Marcus, May 15 2016
a(25) > 4*10^12. - Giovanni Resta, Feb 26 2020

G. L. Cohen and H. J. J. te Riele, Iterating the sum-of-divisors function, Experimental Mathematics, 5 (1996), pp. 93-100.

Cf. A000668, A019276, A019278, A019279, A019281, A019282, A019283, A019284, A019285, A019286, A019287, A019288, A019289, A019290, A019291.

isok(n) = sigma(sigma(n))/n == 8; \\ Michel Marcus, May 15 2016

a(19) from Jud McCranie, Nov 13 2001
a(20)-a(21) from Jud McCranie, Jan 29 2012
a(22)-a(24) from Giovanni Resta, Feb 26 2020

A019286 Let sigma_m (n) be result of applying sum-of-divisors function m times to n; call n (m,k)-perfect if sigma_m (n) = k*n; sequence gives the (2,9)-perfect numbers.

Original entry on oeis.org

168, 10752, 331520, 691200, 1556480, 1612800, 106151936, 5099962368, 4010593484800

Offset: 1

N. J. A. Sloane

See also the Cohen-te Riele links under A019276.
No other terms < 5*10^11. - Jud McCranie, Feb 08 2012
4010593484800 is also a term. See comment in A019278. - Michel Marcus, May 15 2016

Graeme L. Cohen and Herman J. J. te Riele, Iterating the sum-of-divisors function, Experimental Mathematics, 5 (1996), pp. 93-100.

Cf. A000668, A019278, A019279, A019281, A019282, A019283, A019284, A019285, A019286, A019287, A019288, A019289, A019290, A019291.

isok(n) = sigma(sigma(n))/n  == 9; \\ Michel Marcus, May 12 2016

a(8) by Jud McCranie, Jan 28 2012

a(9) from Giovanni Resta, Feb 26 2020

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

Original entry on oeis.org

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

A135653 Divisors of 496 (the 3rd perfect number), written in base 2.

Original entry on oeis.org

1, 10, 100, 1000, 10000, 11111, 111110, 1111100, 11111000, 111110000

Offset: 1

Omar E. Pol, Feb 23 2008, Mar 03 2008

The number of divisors of the third perfect number is equal to 2*A000043(3)=A061645(3)=10.

			The structure of divisors of 496 (see A018487)
-------------------------------------------------------------------------
n ... Divisor . Formula ....... Divisor written in base 2 ...............
-------------------------------------------------------------------------
1)......... 1 = 2^0 ........... 1
2)......... 2 = 2^1 ........... 10
3)......... 4 = 2^2 ........... 100
4)......... 8 = 2^3 ........... 1000
5)........ 16 = 2^4 ........... 10000 ... (The 3rd superperfect number)
6)........ 31 = 2^5 - 2^0 ..... 11111 ... (The 3rd Mersenne prime)
7)........ 62 = 2^6 - 2^1 ..... 111110
8)....... 124 = 2^7 - 2^2 ..... 1111100
9)....... 248 = 2^8 - 2^3 ..... 11111000
10)...... 496 = 2^9 - 2^4 ..... 111110000 ... (The 3rd perfect number)

Index entries for sequences related to divisors of numbers

For more information see A018487 (Divisors of 496). Cf. A000043, A000079, A000396, A000668, A019279, A061645, A061652.

FromDigits[IntegerDigits[#,2]]&/@Divisors[496] (* Harvey P. Dale, Dec 02 2018 *)

apply(n->fromdigits(binary(n)), divisors(496)) \\ Charles R Greathouse IV, Jun 21 2017

a(n)=A018487(n), written in base 2. Also, for n=1 .. 10: If n<=(A000043(3)=5) then a(n) is the concatenation of the digit "1" and n-1 digits "0" else a(n) is the concatenation of A000043(3)=5 digits "1" and (n-1-A000043(3)) digits "0".

A139234 First differences of even superperfect numbers A061652.

Original entry on oeis.org

2, 12, 48, 4032, 61440, 196608, 1073479680, 1152921503533105152, 309485008668423564117934080, 81129328929596860350720280363008, 85070510600596201259161956068936908800

Offset: 1

Omar E. Pol, Apr 18 2008

nonn

First differences of Mersenne primes A000668, divided by 2 (see A139231).

Also, first differences of superperfect numbers A019279, if there are no odd superperfect numbers.

			a(2) = 12 because A061652(2) = 4 and A061652(3) = 16 then 16 - 4 = 12.

Cf. A000668, A019279, A061652, A139228, A139229, A139230, A139231, A139232, A139233, A139235, A139236, A139237.

Differences[Table[2^(MersennePrimeExponent[n]-1),{n,12}]] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Aug 18 2020 *)

a(n) = A061652(n+1) - A061652(n) = A139231(n)/2. Also, a(n) = A019279(n+1) - A019279(n), if there are no odd superperfect numbers.

a(8)-a(11) from A139231(n)/2 by Jinyuan Wang, Mar 04 2020

A019287 Let sigma_m (n) be result of applying sum-of-divisors function m times to n; call n (m,k)-perfect if sigma_m (n) = k*n; sequence gives the (2,10)-perfect numbers.

Original entry on oeis.org

480, 504, 13824, 32256, 32736, 1980342, 1396617984, 3258775296, 14763499520, 38385098752

Offset: 1

N. J. A. Sloane

See also the Cohen-te Riele links under A019276.
No other terms < 5*10^11. - Jud McCranie, Feb 08 2012
a(11) > 4*10^12, if it exists. - Giovanni Resta, Feb 26 2020

G. L. Cohen and H. J. J. te Riele, Iterating the sum-of-divisors function, Experimental Mathematics, 5 (1996), pp. 93-100.

Cf. A019278, A019279, A019281, A019282, A019283, A019284, A019285, A019286, A019288, A019289, A019290, A019291.

More terms from Jud McCranie, Nov 13 2001; a(9) Jan 29 2012, a(10) Feb 08 2012

A019288 Let sigma_m (n) be result of applying sum-of-divisors function m times to n; call n (m,k)-perfect if sigma_m (n) = k*n; sequence gives the (2,11)-perfect numbers.

Original entry on oeis.org

4404480, 57669920, 238608384

Offset: 1

N. J. A. Sloane

See also the Cohen-te Riele links under A019276.
No other terms < 5*10^11. - Jud McCranie, Feb 08 2012
a(4) > 4*10^12. - Giovanni Resta, Feb 26 2020
53283599155200, 2914255525994496 and 3887055949004800 are also terms. - Michel Marcus, Feb 27 2020

Graeme L. Cohen and Herman J. J. te Riele, Iterating the sum-of-divisors function, Experimental Mathematics, 5 (1996), pp. 93-100.

Cf. A019278, A019279, A019281, A019282, A019283, A019284, A019285, A019286, A019287, A019289, A019290, A019291.

isok(n) = sigma(sigma(n))/n  == 11; \\ Michel Marcus, Feb 27 2020

A019289 Let sigma_m (n) be result of applying sum-of-divisors function m times to n; call n (m,k)-perfect if sigma_m (n) = k*n; sequence gives the (2,12)-perfect numbers.

Original entry on oeis.org

2200380, 8801520, 14913024, 35206080, 140896000, 459818240, 775898880, 2253189120, 16785793024, 22648550400, 36051025920, 51001180160, 144204103680

Offset: 1

N. J. A. Sloane

See also the Cohen-te Riele links under A019276.
No others < 5*10^11. - Jud McCranie, Feb 08 2012
a(14) > 4*10^12. - Giovanni Resta, Feb 26 2020
6640556211576, 82863343951872, 182140970374656, 480965999895576, 590660008673280, 886341160140800, 5562693163417600, 9386507580211200 are also terms. - Michel Marcus, Feb 27 2020

Graeme L. Cohen and Herman J. J. te Riele, Iterating the sum-of-divisors function, Experimental Mathematics, 5 (1996), pp. 93-100.

Cf. A019278, A019279, A019281, A019282, A019283, A019284, A019285, A019286, A019287, A019288, A019290, A019291.

isok(n) = sigma(sigma(n))/n  == 12; \\ Michel Marcus, Feb 27 2020

More terms from Jud McCranie, Nov 13 2001, a(9) Feb 01 2012, a(10)-a(13) on Feb 08 2012

A019290 Let sigma_m (n) be result of applying sum-of-divisors function m times to n; call n (m,k)-perfect if sigma_m (n) = k*n; sequence gives the (2,13)-perfect numbers.

Original entry on oeis.org

57120, 932064, 3932040, 251650560

Offset: 1

N. J. A. Sloane

See also the Cohen-te Riele links under A019276.
No other terms < 5*10^11. - Jud McCranie, Feb 08 2012
11383810648416 is also a term. See comment in A019278. - Michel Marcus, May 15 2016
a(5) > 4*10^12. - Giovanni Resta, Feb 26 2020
50248050278400, 117245450649600, 86575337046016000 are also terms. - Michel Marcus, Feb 27 2020

Graeme L. Cohen and Herman J. J. te Riele, Iterating the sum-of-divisors function, Experimental Mathematics, 5 (1996), pp. 93-100.

Cf. A019276, A019278, A019279, A019281, A019282, A019283, A019284, A019285, A019286, A019287, A019288, A019289, A019291.

isok(n) = sigma(sigma(n))/n == 13; \\ Michel Marcus, May 15 2016

A019291 Let sigma_m (n) be result of applying sum-of-divisors function m times to n; call n (m,k)-perfect if sigma_m (n) = k*n; sequence gives the (2,14)-perfect numbers.

Original entry on oeis.org

217728, 1278720, 2983680, 5621760, 14008320, 298721280, 955367424, 1874780160, 4874428416, 1957928934528

Offset: 1

N. J. A. Sloane

nonn

See also the Cohen-te Riele links under A019276.
No other terms < 5*10^11. - Jud McCranie, Feb 08 2012
36095341363200 is also a term. See comment in A019278. - Michel Marcus, May 15 2016
a(11) > 4*10^12. - Giovanni Resta, Feb 26 2020

Graeme L. Cohen and Herman J. J. te Riele, Iterating the sum-of-divisors function, Experimental Mathematics, 5 (1996), pp. 93-100.

Cf. A019276, A019278, A019279, A019281, A019282, A019283, A019284, A019285, A019286, A019287, A019288, A019289, A019290.

isok(n) = sigma(sigma(n))/n == 14; \\ Michel Marcus, May 15 2016

More terms from Jud McCranie, Nov 13 2001
a(9) from Jud McCranie, Jan 28 2012
a(10) from Giovanni Resta, Feb 26 2020

cp's OEIS Frontend

A019285 Let sigma_m (n) be result of applying sum-of-divisors function m times to n; call n (m,k)-perfect if sigma_m (n) = k*n; sequence gives the (2,8)-perfect numbers.

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A019286 Let sigma_m (n) be result of applying sum-of-divisors function m times to n; call n (m,k)-perfect if sigma_m (n) = k*n; sequence gives the (2,9)-perfect numbers.

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

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A135653 Divisors of 496 (the 3rd perfect number), written in base 2.

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Mathematica

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A139234 First differences of even superperfect numbers A061652.

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A019287 Let sigma_m (n) be result of applying sum-of-divisors function m times to n; call n (m,k)-perfect if sigma_m (n) = k*n; sequence gives the (2,10)-perfect numbers.

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A019288 Let sigma_m (n) be result of applying sum-of-divisors function m times to n; call n (m,k)-perfect if sigma_m (n) = k*n; sequence gives the (2,11)-perfect numbers.

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A019289 Let sigma_m (n) be result of applying sum-of-divisors function m times to n; call n (m,k)-perfect if sigma_m (n) = k*n; sequence gives the (2,12)-perfect numbers.

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