A019276 -id:A019276 - cp's OEIS frontend

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.

4404480, 57669920, 238608384

Offset: 1

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

A019295 a(n) = sigma(sigma(...(sigma(n))...)) / n, where sigma (A000203) is iterated until a multiple of n is reached.

Original entry on oeis.org

1, 2, 5, 2, 24, 2, 24, 3, 168, 12, 1834560, 10, 84480, 12, 4, 2, 92520, 20, 62720, 84, 3, 49920, 6516224, 7, 881280, 28, 3360, 2, 517517500266693633076805172570524811961093324800, 728, 912, 18, 19767296, 46260, 144, 42, 30349648609280, 38644089120, 30, 663, 34042889727216750428160

Offset: 1

N. J. A. Sloane

nonn

The minimal number of iterations of the sigma function until a multiple of n is reached (after the initial n) is given in A019294.

See also the Cohen-te Riele links in A019276.

Graeme L. Cohen and Herman J. J. te Riele, Iterating the sum-of-divisors function, Experimental Mathematics, 5 (1996), pp. 93-100.
Experimental Mathematics, Volume 5 (1996) ["Outdated Archival Version" as of 2005.]

Cf. A019276 (megaperfect numbers: where A019294 reaches records), A019276 (record values), A019294 (number of iterations needed to reach a multiple of n).

f:=func; a:=[]; for n in [1..41] do k:=n; while f(k) mod n ne 0 do k:=f(k); end while; Append(~a,f(k) div n); end for; a; // Marius A. Burtea, Jan 11 2020

apply( {A019295(n,s=n)=while((s=sigma(s))%n,);s\n}, [1..50]) \\ M. F. Hasler, Jan 08 2020

More terms from Max Alekseyev, Sep 22 2016

Edited by M. F. Hasler, Jan 08 2020

A019293 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 (4,k)-perfect numbers.

Original entry on oeis.org

1, 2, 3, 4, 6, 8, 10, 12, 15, 18, 21, 24, 26, 32, 39, 42, 60, 65, 72, 84, 96, 160, 182, 336, 455, 512, 672, 896, 960, 992, 1023, 1280, 1536, 1848, 2040, 2688, 4092, 5472, 5920, 7808, 7936, 10416, 11934, 16352, 16380, 18720, 20384, 21824, 23424, 24564, 29127, 30240, 33792, 36720, 41440

Offset: 1

N. J. A. Sloane

nonn

Similarly to A019278, 2 and sigma(2) are both terms, and this can happen also for further iterations of sigma on known terms, thus providing new terms outside the current known range. - Michel Marcus, May 01 2017

Also it occurs that many divisors of A019278 are terms of this sequence. - Michel Marcus, May 28 2017

			10 is a term because applying sigma four times we see that 10 -> 18 -> 39 -> 168 -> 120, and 120 = 12*10. So 10 is a (4,12)-perfect number.

Michel Marcus, Table of n, a(n) for n = 1..320
Graeme L. Cohen and Herman J. J. te Riele, Iterating the sum-of-divisors function, Experimental Mathematics, 5 (1996), pp. 93-100.
Michel Marcus, Unexhaustive list of terms

Cf. A019276, A019278, A019292, A129076.

isok(n) = denominator(sigma(sigma(sigma(sigma(n))))/n) == 1; \\ Michel Marcus, Apr 29 2017

Corrected by Michel Marcus, Apr 29 2017

A019277 Records in A019294, number of iterations of the sigma function to reach a multiple of the starting value.

Original entry on oeis.org

1, 2, 4, 5, 7, 15, 16, 17, 78, 97, 101, 120, 174, 214, 239, 261, 263, 296, 380, 557, 1287, 1524, 1722, 1911, 2023, 2373

Offset: 1

N. J. A. Sloane

Original name: Let sigma_m(n) be the result of applying the sum-of-divisors function m times to n; let m(n) = min m such that n divides sigma_m (n); let k(n) = sigma_{m(n)}(n)/n; sequence gives k(n) for the megaperfect numbers n, where m(n) increases.
Records in A019294. a(n>=23) depend on a few probable primes.
See also the Cohen-te Riele links under A019276.
The original name mentioned the sequence of ratios k, i.e., A019295(A019276) = (1, 2, 5, 24, 168, 1834560, 6516224, 881280, ...), at present not listed in the OEIS. - M. F. Hasler, Jan 07 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 (megaperfect numbers: where A019294 has records), A019294 (min m: n|sigma^m(n)), A019295 (sigma^m(n)/n with m = A019294).

f[n_, m_] := Block[{d = DivisorSigma[1, n]}, If[Mod[d, m] == 0, 0, d]]; g[n_] := Length[ NestWhileList[ f[ #, n] &, n, # != 0 &]] - 1; a = 0; Do[b = g[n]; If[b > a, a = b; Print[ a]], {n, 460}] (* Robert G. Wilson v, Jun 24 2005 *)

{M=0; for(n=1,oo, my(s=n,m=1); while((s=sigma(s))%n,m++); m>M&&print1(M=m,","))} \\ M. F. Hasler, Jan 07 2020

a(n) = A019294(A019276(n)). - M. F. Hasler, Jan 07 2020

Definition corrected by M. F. Hasler, Jan 07 2020

A019280 Let sigma_m(n) be result of applying the sum-of-divisors function m times to n; call n (m,k)-perfect if sigma_m(n) = k*n; sequence gives log_2 of the (2,2)-perfect numbers.

Original entry on oeis.org

1, 2, 4, 6, 12, 16, 18, 30, 60

Offset: 1

N. J. A. Sloane

Cohen and te Riele prove that any even (2,2)-perfect number (a "superperfect" number) must be of the form 2^(p-1) with 2^p-1 prime (Suryanarayana) and the converse also holds. Any odd superperfect number must be a perfect square (Kanold). Searches up to > 10^20 did not find any odd examples. - Ralf Stephan, Jan 16 2003

See also the Cohen-te Riele links under A019276.

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.

Coincides with A000043(n) - 1 unless odd superperfect numbers exist.

a(8)-a(9) from Jud McCranie, Jun 01 2000

cp's OEIS Frontend

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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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.

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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.

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A019295 a(n) = sigma(sigma(...(sigma(n))...)) / n, where sigma (A000203) is iterated until a multiple of n is reached.

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A019293 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 (4,k)-perfect numbers.

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A019277 Records in A019294, number of iterations of the sigma function to reach a multiple of the starting value.

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A019280 Let sigma_m(n) be result of applying the sum-of-divisors function m times to n; call n (m,k)-perfect if sigma_m(n) = k*n; sequence gives log_2 of the (2,2)-perfect numbers.

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