A285900 Sum of all parts of all partitions of all positive integers <= n into consecutive parts.

1, 3, 9, 13, 23, 35, 49, 57, 84, 104, 126, 150, 176, 204, 264, 280, 314, 368, 406, 446, 530, 574, 620, 668, 743, 795, 903, 959, 1017, 1137, 1199, 1231, 1363, 1431, 1571, 1679, 1753, 1829, 1985, 2065, 2147, 2315, 2401, 2489, 2759, 2851, 2945, 3041, 3188, 3338, 3542, 3646, 3752, 3968, 4188, 4300, 4528, 4644, 4762, 5002

Offset

1,2

Comments

a(n) is also the sum of all parts of all partitions of all positive integers <= n into an odd number of equal parts. - Omar E. Pol, Jun 05 2017

Links

G. C. Greubel, Table of n, a(n) for n = 1..5000

Formula

a(n) = Sum_{k=1..floor((n+1)/2)} (2*k-1)/2 * floor(n/(2*k-1)) * floor(1 + n/(2*k-1)). - Daniel Suteu, Nov 06 2018

Example

For n = 15, there are four partitions of 15 into consecutive parts: [15], [8, 7], [6, 5, 4] and [5, 4, 3, 2, 1]. The sum of all parts of these four partitions is 60, and a(14) = 204, so a(15) = 204 + 60 = 264.

Mathematica

a285900[n_] := Accumulate[Map[# DivisorSum[#, 1 &, OddQ] &, Range[n]]]
a285900[60] (* data *) (* Hartmut F. W. Hoft, Jun 06 2017 *)

Prog

(PARI) a(n)=sum(i=1, n, i * sumdiv(i, d, d%2)); \\ Andrew Howroyd, Nov 06 2018
(PARI) a(n)=sum(k=1, (n+1)\2, (2*k - 1)/2 * (n\(2*k - 1)) * (1 + n\(2*k - 1))); \\ Andrew Howroyd, Nov 06 2018

Crossrefs

Partial sums of A245579.

Cf. A001227, A060831, A204217, A237593, A285891, A285899.

Sequence in context: A004617 A135370 A206802 * A022408 A126827 A363628

Adjacent sequences: A285897 A285898 A285899 * A285901 A285902 A285903

Keyword

nonn

Author

Omar E. Pol, May 02 2017

Status

approved