cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-10 of 13 results. Next

A019281 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 (2,3)-perfect numbers.

Original entry on oeis.org

8, 21, 512
Offset: 1

Views

Author

N. J. A. Sloane

Keywords

Comments

See also the Cohen-te Riele links under A019276.
No further term < 10^9 [see Table 1].
No other terms < 5*10^11. - Jud McCranie, Feb 08 2012
a(4) > 4*10^12, if it exists. - Giovanni Resta, Feb 26 2020

Links

Crossrefs

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

A019282 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 (2,4)-perfect numbers.

Original entry on oeis.org

15, 1023, 29127, 355744082763
Offset: 1

Views

Author

N. J. A. Sloane

Keywords

Comments

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

Links

Crossrefs

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

Programs

  • Mathematica
    Select[Range[100000], DivisorSigma[1, DivisorSigma[1, #]]/# == 4 &] (* Robert Price, Apr 07 2019 *)
  • PARI
    isok(n) = sigma(sigma(n))/n  == 4; \\ Michel Marcus, May 12 2016

Extensions

a(4) from Jud McCranie, Feb 08 2012

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.

Original entry on oeis.org

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

Views

Author

N. J. A. Sloane

Keywords

Comments

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

Links

Crossrefs

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

Programs

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

Extensions

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

Views

Author

N. J. A. Sloane

Keywords

Comments

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

Links

Crossrefs

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

Programs

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

Extensions

a(8) by Jud McCranie, Jan 28 2012
a(9) from Giovanni Resta, Feb 26 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

Views

Author

N. J. A. Sloane

Keywords

Comments

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

Links

Crossrefs

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

Extensions

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

Views

Author

N. J. A. Sloane

Keywords

Comments

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

Links

Crossrefs

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

Programs

  • PARI
    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

Views

Author

N. J. A. Sloane

Keywords

Comments

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

Links

Crossrefs

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

Programs

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

Extensions

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

Views

Author

N. J. A. Sloane

Keywords

Comments

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

Links

Crossrefs

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

Programs

  • PARI
    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

Views

Author

N. J. A. Sloane

Keywords

Comments

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

Links

Crossrefs

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

Programs

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

Extensions

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

A173430 Last of consecutive coprime iterations of sum-of-divisors function.

Original entry on oeis.org

1, 15, 15, 15, 6, 6, 15, 15, 14, 10, 12, 12, 14, 14, 15, 104, 18, 18, 20, 20, 104, 22, 24, 24, 104, 26, 40, 28, 30, 30, 104, 104, 33, 34, 48, 91, 38, 38, 56, 40, 42, 42, 44, 44, 45, 46, 48, 48, 80, 255, 51, 52, 54, 54, 72, 56, 80, 58, 60, 60, 62, 62, 104, 255, 84, 66, 68, 68
Offset: 1

Views

Author

Walter Nissen, Feb 18 2010

Keywords

Examples

			Calculating sum-of-divisors ( ... sum-of-divisors ( sum-of-divisors ( 4 ) ) ... ) the iterates are 4, 7, 8, 15, 24, ... .
The initial, consecutive, pairwise, coprime iterates are 4, 7, 8, 15, so a(4) = 15 .
Here sigma ( 4 ) = 7, sigma ( sigma ( 4 ) ) = sigma ( 7 ) = 8, etc.

References

  • Oystein Ore, Number Theory and Its History, 1988, Dover Publications, ISBN 0486656209, pp. 88-96.

Links

Crossrefs

Cf. A129246 and the references there, A019294, A019295, A000203, A051027, A019284, A019277.

Programs

  • Mathematica
    a[1] = 1; a[n_] := Module[{k = n}, While[CoprimeQ[k, (s = DivisorSigma[1, k])], k = s]; k]; Array[a, 68] (* Amiram Eldar, Sep 02 2019 *)
Showing 1-10 of 13 results. Next

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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