A285900 - OEIS

%I #56 Nov 07 2018 03:54:00

%S 1,3,9,13,23,35,49,57,84,104,126,150,176,204,264,280,314,368,406,446,

%T 530,574,620,668,743,795,903,959,1017,1137,1199,1231,1363,1431,1571,

%U 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

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

%C 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

%H G. C. Greubel, <a href="/A285900/b285900.txt">Table of n, a(n) for n = 1..5000</a>

%F 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

%e 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.

%t a285900[n_] := Accumulate[Map[# DivisorSum[#, 1 &, OddQ] &, Range[n]]]

%t a285900[60] (* data *) (* _Hartmut F. W. Hoft_, Jun 06 2017 *)

%o (PARI) a(n)=sum(i=1, n, i * sumdiv(i, d, d%2)); \\ _Andrew Howroyd_, Nov 06 2018

%o (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

%Y Partial sums of A245579.

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

%K nonn

%O 1,2

%A _Omar E. Pol_, May 02 2017