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 1-3 of 3 results.

A336696 Sum of odd divisors of 1+sigma(n).

Original entry on oeis.org

1, 1, 6, 1, 8, 14, 13, 1, 8, 20, 14, 30, 24, 31, 31, 1, 20, 6, 32, 44, 48, 38, 31, 62, 1, 44, 42, 80, 32, 74, 48, 1, 57, 72, 57, 24, 56, 62, 80, 112, 44, 98, 78, 108, 80, 74, 57, 156, 30, 48, 74, 156, 72, 133, 74, 133, 121, 112, 62, 183, 104, 98, 192, 1, 108, 180, 96, 128, 98, 180, 74, 57, 124, 144, 156, 192, 98
Offset: 1

Views

Author

Antti Karttunen, Jul 31 2020

Keywords

Links

Crossrefs

Cf. A000203, A000593, A088580, A193337, A324294, A332459, A336691, A336692, A336693, A336694, A336696.

Programs

  • Mathematica
    Table[Total[Select[Divisors[1+DivisorSigma[1,n]],OddQ]],{n,80}] (* Harvey P. Dale, Jan 01 2022 *)
  • PARI
    A000593(n) = sigma(n>>valuation(n, 2));
A336696(n) = A000593(1+sigma(n));

Formula

a(n) = A000593(1+A000203(n)) = A000593(A088580(n)) = A000593(A332459(n)).

A193336 Sum of even divisors of sigma(n).

Original entry on oeis.org

0, 0, 6, 0, 8, 24, 14, 0, 0, 26, 24, 48, 16, 56, 56, 0, 26, 0, 36, 64, 62, 78, 56, 144, 0, 64, 84, 112, 48, 182, 62, 0, 120, 80, 120, 0, 40, 144, 112, 156, 64, 248, 72, 192, 112, 182, 120, 192, 0, 0, 182, 114, 80, 336, 182, 336, 180, 156, 144, 448, 64, 248, 196, 0, 192, 390, 108, 208, 248, 390, 182, 0, 76, 160, 192, 288, 248, 448, 180, 256, 0
Offset: 1

Views

Author

Michel Lagneau, Jul 23 2011

Keywords

Comments

sigma(n) = sum of divisors of n: A000203 (also called sigma_1(n)).

Examples

			a(14) = 56 because sigma(14) = 24 and the sum of the 6 even divisors {2, 4, 6, 8, 12, 24} is 56.

Links

Crossrefs

Cf. A000203, A028982, A051027, A146076, A193334, A193337.

Programs

  • Mathematica
    Table[Total[Select[Divisors[DivisorSigma[1,n]], EvenQ[ # ]&]], {n, 53}]
  • PARI
    A193336(n) = { my(s=sigma(n)); sumdiv(s,d,(!(d%2))*d); }; \\ Antti Karttunen, Nov 18 2017

Formula

a(n) + A193337(n) = A051027(n). - Antti Karttunen, Nov 18 2017
From Amiram Eldar, Mar 30 2024: (Start)
a(n) = A146076(A000203(n)).
a(n) = 0 if and only if n is in A028982. (End)

Extensions

More terms from Antti Karttunen, Nov 18 2017

A252540 Numbers k such that A000593(A146076(k)) = k. (A000593(n) is the sum of the odd divisors of n; A146076(n) is the sum of the even divisors of n.)

Original entry on oeis.org

4, 8, 32, 128, 320, 8192, 131072, 524288, 11243520, 2147483648
Offset: 1

Views

Author

Michel Lagneau, Dec 18 2014

Keywords

Comments

All integers of the form 2^p where 2^p-1 is prime are terms (see A000668). The terms that are not of this form are 320, 11243520. Are there any other? [Edited by Michel Marcus, Nov 22 2022]
a(10) > 10^9. - Michel Marcus, Jan 06 2015
a(11) > 10^10. - Michel Marcus, Nov 22 2022
All terms are even, because odd numbers will fail with A146076(odd) = 0. - Michel Marcus, Nov 22 2022
Numbers k such that A193337(k/2) = k. - Michel Marcus, Nov 23 2022

Examples

			The divisors of 8 are {1, 2, 4, 8}, so the sum of even divisors of 8 is 2 + 4 + 8 = 14, and the divisors of 14 are {1, 2, 7, 14}, so the sum of odd divisors of 14 is 1 + 7 = 8; thus, 8 is in the sequence. That is A000593(A146076(8)) = A000593(14) = 8.

Crossrefs

Cf. A000593, A146076, A000668, A193337, A252541.

Programs

  • Mathematica
    f[n_]:= Plus @@ Select[ Divisors@ n, OddQ];g[n_]:= Plus @@ Select[ Divisors@ n, EvenQ];Do[If[f[g[n]]==n,Print[n]],{n,1,10^8}]
  • PARI
    sod(n) = if (n, sigma(n>>valuation(n, 2)), 0); \\ A000593
sed(n) = if (n%2, 0, 2*sigma(n/2)); \\ A146076
isok(n) = sod(sed(n)) == n;
lista(nn) = forstep(n=2, nn, 2, if(isok(n), print1(n, ", "))); \\ Michel Marcus, Nov 22 2022

Extensions

a(10) from Michel Marcus, Nov 22 2022
Showing 1-3 of 3 results.

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