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 1-7 of 7 results.

A129246 Iterated sum of divisors array A[k,n] = k-th iterate of sigma(n), by upward antidiagonals.

Original entry on oeis.org

1, 1, 3, 1, 4, 4, 1, 7, 7, 7, 1, 8, 8, 8, 6, 1, 15, 15, 15, 12, 12, 1, 24, 24, 24, 28, 28, 8, 1, 60, 60, 60, 56, 56, 15, 15, 1, 168, 168, 168, 120, 120, 24, 24, 13, 1, 480, 480, 480, 360, 360, 60, 60, 14, 18, 1, 1512, 1512, 1512, 1170, 1170, 168, 168, 24, 39, 12, 1, 4800, 4800
Offset: 1

Views

Author

Jonathan Vos Post, May 27 2007

Keywords

Examples

			Array begins:
k / sigma(...sigma(n)..) nested k deep.
1.|.1...3...4....7....6....12....8....15...13....18...
2.|.1...4...7....8...12....28...15....24...14....39...
3.|.1...7...8...15...28....56...24....60...24....56...
4.|.1...8..15...24...56...120...60...168...60...120...
5.|.1..15..24...60..120...360..168...480..168...360...
6.|.1..24..60..168..360..1170..480..1512..480..1170...
7.|.1..60.168..480.1170..3276.1512..4800.1512..3276...
8.|.1.168.480.1512.3276.10192.4800.15748.4800.10192...

Links

Crossrefs

Cf. A000203 (row 1), A051027 (row 2), A066971 (row 3).
Cf. A000012 (column 1), A007497 (column 2), A090896 (main diagonal).

Programs

  • Maple
    A129246 := proc(k,n) option remember ; if k= 1 then numtheory[sigma](n); else A129246(k-1,numtheory[sigma](n)) ; fi ; end: for d from 1 to 13 do for n from 1 to d do printf("%d, ",A129246(d+1-n,n)) ; od: od: # R. J. Mathar, Oct 09 2007
  • Mathematica
    T[n_, k_] := T[n, k] = If[n == 1, DivisorSigma[1, k], DivisorSigma[1, T[n-1, k]]];
Table[T[d-k+1, k], {d, 1, 13}, {k, 1, d}] // Flatten (* Jean-François Alcover, Sep 23 2022, after R. J. Mathar, except that T(n,k) replaces the unusual A(k,n) *)

Formula

A[k,n] = sigma^k(n), where sigma^k denotes functional iteration.

Extensions

More terms from R. J. Mathar, Oct 09 2007

A129076 a(n) = sigma(sigma(sigma(sigma(n)))), where sigma(n) = sum of divisors of n.

Original entry on oeis.org

1, 8, 15, 24, 56, 120, 60, 168, 60, 120, 120, 360, 168, 480, 480, 104, 120, 360, 252, 728, 210, 248, 480, 1512, 104, 728, 546, 1170, 336, 992, 210, 576, 504, 1170, 504, 480, 480, 1512, 1170, 1344, 728, 1680, 504, 1560, 1512, 992, 504, 1560, 384, 432, 992, 588
Offset: 1

Views

Author

Jonathan Vos Post, May 27 2007

Keywords

Comments

a(n) = A000203(A000203(A051027(n))) = A051027(A000203(A000203(n))) = A000203(A051027(A000203(n))) = A000203(A066971(n)) = A066971(A000203(n)) = A051027(A051027(n)). - Jaroslav Krizek, Jul 19 2009

Links

Crossrefs

Cf. A000203, A051027, A066971.
Cf. A019293.

Programs

  • Magma
    [ SumOfDivisors(SumOfDivisors(SumOfDivisors(SumOfDivisors(n)))) : n in [1..100]];
  • Mathematica
    Nest[DivisorSigma[1,#]&,Range[60],4] (* Harvey P. Dale, Oct 05 2011 *)
  • PARI
    a(n) = sigma(sigma(sigma(sigma(n)))); \\ Michel Marcus, Apr 29 2017

Formula

a(n) = A000203(A000203(A000203(A000203(n)))).

A354197 a(n) = A064989(sigma(sigma(sigma(A003961(n))))), where A003961 shifts the prime factorization one step towards larger primes, and A064989 shifts it back towards smaller primes.

Original entry on oeis.org

1, 1, 5, 2, 2, 10, 5, 44, 20, 11, 6, 6, 5, 5, 5, 3, 2, 20, 10, 4, 10, 12, 66, 6, 58, 10, 204, 204, 11, 5, 10, 986, 20, 2, 55, 113, 20, 55, 12, 2, 5, 55, 5, 29, 40, 132, 12, 15, 40, 58, 132, 10, 6, 6, 6, 18, 5, 8, 20, 6, 22, 145, 78, 262, 5, 20, 10, 170, 10, 40, 6, 2486, 2, 40, 50, 12, 40, 12, 20, 6, 60, 5, 110, 20
Offset: 1

Views

Author

Antti Karttunen, May 24 2022

Keywords

Comments

For any hypothetical odd perfect number opn that is not a multiple of 3, it holds that a(n) = A354195(n) = 2*n, where n = A064989(opn).

Links

Crossrefs

Cf. A000203, A003961, A064989, A066971, A354198 [= A064989(a(A003961(n)))].
Cf. also A326042, A354195.

Programs

  • PARI
    A003961(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
A064989(n) = { my(f=factor(n>>valuation(n, 2))); for(i=1, #f~, f[i, 1] = precprime(f[i, 1]-1)); factorback(f); };
A354197(n) = A064989(sigma(sigma(sigma(A003961(n)))));

Formula

a(n) = A064989(A066971(A003961(n))).

A354198 a(n) = A064989(A064989(sigma(sigma(sigma(A003961(A003961(n))))))), where A003961 shifts the prime factorization of n one step towards larger primes, and A064989 shifts it back towards smaller primes.

Original entry on oeis.org

1, 3, 1, 3, 3, 3, 2, 26, 23, 3, 3, 3, 1, 3, 21, 6, 3, 9, 14, 22, 2, 2, 7, 182, 3, 14, 313, 201, 3, 3, 3, 603, 3, 3, 3, 115, 3, 3, 2, 3, 3, 21, 2, 9, 9, 3, 2, 75, 2, 22, 3, 109, 3, 21, 46, 109, 2, 23, 7, 154, 3, 6, 22, 222, 2, 14, 2, 22, 29, 6, 1, 78, 3, 161, 69, 1407, 6, 2, 21, 44, 7, 21, 14, 201, 21, 39, 3, 529
Offset: 1

Views

Author

Antti Karttunen, May 24 2022

Keywords

Comments

For any hypothetical odd perfect number opn that is not a multiple of 3, it holds that a(n) = A354196(n) = A348750(n) = n, where n = A064989(A064989(opn)). See also comments in A353365.

Links

Crossrefs

Cf. A000203, A003961, A064989, A066971, A326042, A353365, A354197.
Cf. also A348750, A354196.

Programs

  • PARI
    A003961(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
A064989(n) = { my(f=factor(n>>valuation(n, 2))); for(i=1, #f~, f[i, 1] = precprime(f[i, 1]-1)); factorback(f); };
A354198(n) = A064989(A064989(sigma(sigma(sigma(A003961(A003961(n)))))));

Formula

a(n) = A064989(A354197(A003961(n))) = A064989(A064989(A066971(A003961(A003961(n))))).

A067065 Numbers k such that sigma(sigma(sigma(k))) == 6*sigma(k).

Original entry on oeis.org

20, 26, 41, 44, 65, 83, 132, 133, 140, 182, 188, 195, 249, 287, 299, 420, 546, 564, 620, 644, 764, 806, 861, 897, 1001, 1115, 1169, 1271, 28644, 32172, 35052, 39116, 40796, 41478, 42315, 47492, 50162, 51513, 52143, 53745, 54033, 54483, 56427, 56642
Offset: 1

Views

Author

Benoit Cloitre, Feb 17 2002

Keywords

Links

Crossrefs

Cf. A000203 (sigma), A066971.

Programs

  • Mathematica
    Select[Range[60000],Nest[DivisorSigma[1,#]&,#,3]==6*DivisorSigma[1,#]&] (* Harvey P. Dale, Oct 04 2016 *)
  • PARI
    isok(k) = { sigma(sigma(sigma(k))) == 6*sigma(k) } \\ Harry J. Smith, May 03 2010

Extensions

Definition corrected and more terms added by Harry J. Smith, May 03 2010

A162964 a(n) = sigma(sigma(sigma(sigma(sigma(n))))).

Original entry on oeis.org

1, 15, 24, 60, 120, 360, 168, 480, 168, 360, 360, 1170, 480, 1512, 1512, 210, 360, 1170, 728, 1680, 576, 480, 1512, 4800, 210, 1680, 1344, 3276, 992, 2016, 576, 1651, 1560, 3276, 1560, 1512, 1512, 4800, 3276, 4064, 1680, 5952, 1560, 5040, 4800, 2016, 1560, 5040
Offset: 1

Views

Author

Jaroslav Krizek, Jul 19 2009

Keywords

Links

Crossrefs

Cf. A000203, A129076, A066971, A051027.
Cf. also A049107.

Programs

  • Maple
    with(numtheory); f:=n->sigma(sigma(sigma(sigma(sigma(n))))); [seq(f(n),n=1..100)];
  • Mathematica
    Table[Nest[DivisorSigma[1,#]&,n,5],{n,50}] (* Harvey P. Dale, Apr 19 2013 *)
  • PARI
    A162964(n) = sigma(sigma(sigma(sigma(sigma(n))))); \\ Antti Karttunen, Nov 18 2017

Formula

a(n) = A000203(A000203(A000203(A000203(A000203(n))))).

Extensions

More terms from N. J. A. Sloane, Mar 20 2010

A292369 Numbers n such that f(f(f(n))) = f(f(n)) + f(n) where f = A000203.

Original entry on oeis.org

2, 4, 16, 25, 64, 4096, 65536, 262144, 1073741824
Offset: 1

Views

Author

Altug Alkan, Sep 15 2017

Keywords

Comments

Numbers n such that A066971(n) = A051027(n) + A000203(n).
A061652 is a subsequence.
Are there any odd terms other than 25?

Examples

			25 = 5^2 is a term because sigma(sigma(sigma(5^2))) = sigma(2^5) = sigma(sigma(5^2)) + sigma(5^2).

Crossrefs

Cf. A000203, A019279, A051027, A061652, A066971.

Programs

  • Mathematica
    f[n_] := DivisorSigma[1, n]; fQ[n_] := f[f[f[n]]] == f[f[n]] + f[n]; Select[ Range@1000000, fQ] (* Robert G. Wilson v, Sep 23 2017 *)
  • PARI
    s(n) = sigma(n);
isok(n) = s(s(s(n)))==s(s(n))+s(n);

Extensions

a(9) from Giovanni Resta, Sep 15 2017
Showing 1-7 of 7 results.

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

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