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 1-5 of 5 results.

A280098 The sum of the divisors of 24*n - 1, divided by 24.

Original entry on oeis.org

1, 2, 3, 5, 6, 7, 7, 8, 11, 10, 11, 14, 13, 17, 15, 16, 19, 18, 28, 20, 21, 24, 25, 31, 25, 30, 27, 31, 35, 30, 31, 35, 38, 41, 35, 36, 37, 38, 54, 46, 41, 45, 43, 53, 49, 46, 57, 48, 62, 55, 51, 55, 56, 76, 55, 60, 57, 63, 71, 60, 80, 62, 63, 77, 65, 66, 67
Offset: 1

Views

Author

Michael Somos, Dec 25 2016

Keywords

Comments

Conjecture: only the integers k in {1, 3, 4, 6, 8, 12, 24} have the property that the sum of the divisors of (k*n-1)/k is always an integer. - Robert G. Wilson v, Dec 25 2016
The finite sequence mentioned in the above conjecture gives the sum of the divisors of the partition numbers of the first seven positive integers (cf. A139041). - Omar E. Pol, Dec 25 2016

Examples

			G.f. = x + 2*x^2 + 3*x^3 + 5*x^4 + 6*x^5 + 7*x^6 + 7*x^7 + 8*x^8 + 11*x^9 + ...

Links

Crossrefs

Cf. A000203, A086463, A139041, A183010, A280097.

Programs

  • Mathematica
    a[ n_] := If[ n < 1, 0, DivisorSigma[ 1, 24 n - 1] / 24];
DivisorSigma[1,24*Range[70]-1]/24 (* Harvey P. Dale, Sep 25 2017 *)
  • PARI
    {a(n) = if( n<1, 0, sigma(24*n - 1) / 24)};

Formula

24 * a(n) = sum of the divisors of A183010(n).
a(n) = A280097(n)/24. - Omar E. Pol, Dec 25 2016
Sum_{k=1..n} a(k) = c * n^2 + O(n*log(n)), where c = Pi^2/18 = 0.548311... (A086463). - Amiram Eldar, Mar 28 2024

A272024 Number of partitions of the sum of the divisors of n.

Original entry on oeis.org

1, 3, 5, 15, 11, 77, 22, 176, 101, 385, 77, 3718, 135, 1575, 1575, 6842, 385, 31185, 627, 53174, 8349, 17977, 1575, 966467, 6842, 53174, 37338, 526823, 5604, 5392783, 8349, 1505499, 147273, 386155, 147273, 64112359, 26015, 966467, 526823, 56634173, 53174, 118114304, 75175, 26543660, 12132164, 5392783
Offset: 1

Views

Author

Omar E. Pol, Apr 19 2016

Keywords

Comments

Also number of partitions of the total number of parts in the partitions of n into equal parts.
Note that one of the partitions of the sum of the divisors of n is also the list of divisors of n in decreasing order, see example.

Examples

			For n = 9 the sum of the divisors of 9 is 1 + 3 + 9 = 13 and the number of partitions of 13 is A000041(13) = 101, so a(9) = 101.
Note that one of the 101 partitions of 13 is [9, 3, 1] and it is also the list of divisors of 9 in decreasing order.

Links

Crossrefs

Cf. A000041, A000203, A056538, A058699, A072861, A139041, A272209.

Programs

  • Mathematica
    Table[PartitionsP@ DivisorSigma[1, n], {n, 46}] (* Michael De Vlieger, Apr 19 2016 *)
  • PARI
    a(n) = numbpart(sigma(n)); \\ Michel Marcus, Apr 19 2016

Formula

a(n) = p(sigma(n)) = A000041(A000203(n)).

A139055 Sum of proper divisors of the number of partitions of n.

Original entry on oeis.org

0, 1, 1, 1, 1, 1, 9, 14, 42, 54, 64, 19, 1, 105, 196, 153, 183, 191, 536, 333, 1548, 1014, 257, 1649, 1282, 4284, 3326, 2870, 1483, 7500, 4390, 4419, 7641, 9866, 7461, 1, 5435, 9097, 38511, 50214, 29913, 33874, 41283, 22041, 47954, 109338, 107806, 77175, 61579, 129998
Offset: 1

Views

Author

Omar E. Pol, Apr 16 2008

Keywords

Examples

			a(7) = 9 because the number of partitions of 7 is 15 and the sum of proper divisors of 15 is equal to 1 + 3 + 5 = 9.

Links

Crossrefs

Cf. A000041, A001065, A139041.

Programs

  • Mathematica
    s[n_] := DivisorSigma[1, n] - n; Array[s[PartitionsP[#]] &, 50] (* Amiram Eldar, Jan 07 2020 *)
  • PARI
    a(n) = my(p=numbpart(n)); sigma(p) - p; \\ Michel Marcus, Jan 07 2020

Formula

a(n) = A001065(A000041(n)).

A280101 a(n) = sigma(sigma(p(n))) = sum of the divisors of the sum of the divisors of number of partitions of n.

Original entry on oeis.org

1, 1, 4, 7, 12, 15, 28, 60, 91, 195, 252, 360, 252, 216, 744, 896, 1020, 1512, 1651, 2400, 3048, 7644, 6552, 4800, 6720, 10890, 24384, 19812, 17360, 20160, 45136, 35280, 40320, 54600, 78624, 68400, 27540, 79248, 115200, 219583, 265980, 200312, 268800, 335160
Offset: 0

Views

Author

Omar E. Pol, Dec 25 2016

Keywords

Examples

			For n = 7 the number of partitions of 7 is p(7) = 15, and the sum of the divisors of 15 is sigma(15) = 1 + 3 + 5 + 15 = 24, and the sum of the divisors of 24 is sigma(24) = 1 + 2 + 3 + 4 + 6 + 8 + 12 + 24 = 60, so a(7) = 60.

Links

Crossrefs

Cf. A000041, A000203, A051027, A139041.

Programs

  • Mathematica
    Array[DivisorSigma[1, DivisorSigma[1, PartitionsP[#]]]&, 43, 0] (* Amiram Eldar, Feb 19 2019 *)
  • PARI
    a(n) = sigma(sigma(numbpart(n))); \\ Michel Marcus, Feb 19 2019

Formula

a(n) = A000203(A000203(A000041(n))) = A051027(A000041(n)).
a(n) = A000203(A139041(n)), n >= 1.

A366581 a(n) = phi(p(n)), where phi is Euler's totient function (A000010) and p(n) is the number of partitions of n (A000041).

Original entry on oeis.org

1, 1, 1, 2, 4, 6, 10, 8, 10, 8, 12, 24, 60, 100, 72, 80, 120, 180, 240, 168, 360, 240, 332, 1000, 720, 880, 672, 1008, 1560, 3280, 1864, 3100, 4840, 5544, 4920, 8800, 17976, 16800, 18480, 12960, 10584, 23040, 24160, 37800, 57600, 43440, 34560, 49896, 84144
Offset: 0

Views

Author

Sean A. Irvine, Oct 13 2023

Keywords

Crossrefs

Cf. A000041, A000010, A087175, A139041.

Programs

Formula

a(n) = A000010(A000041(n)).
Showing 1-5 of 5 results.

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