cp's OEIS Frontend

A296966 Sum of all the parts in the partitions of n into two distinct parts such that the smaller part divides the larger.

%I A296966 #22 Feb 23 2018 11:32:39
%S A296966 0,0,3,4,5,12,7,16,18,20,11,48,13,28,45,48,17,72,19,80,63,44,23,144,
%T A296966 50,52,81,112,29,180,31,128,99,68,105,252,37,76,117,240,41,252,43,176,
%U A296966 225,92,47,384,98,200,153,208,53,324,165,336,171,116,59,600,61
%N A296966 Sum of all the parts in the partitions of n into two distinct parts such that the smaller part divides the larger.
%H A296966 <a href="/index/Par#part">Index entries for sequences related to partitions</a>
%F A296966 a(n) = n * Sum_{i=1..floor((n-1)/2)} floor(n/i) - floor((n-1)/i).
%F A296966 a(n) = n * A023645(n). - _Robert G. Wilson v_, Dec 24 2017
%e A296966 From _Wesley Ivan Hurt_, Feb 21 2018: (Start)
%e A296966 a(5) = 5; there is one partition of 5 into two distinct parts such that the smaller part divides the larger, namely (4,1), so the sum of the parts is then 4 + 1 = 5.
%e A296966 a(6) = 12; the partitions of 6 into two distinct parts such that the smaller part divides the larger are (5,1) and (4,2), and the sum of the parts is then 5 + 1 + 4 + 2 = 12.
%e A296966 a(7) = 7; there is one partition of 7 into two distinct parts such that the smaller part divides the larger, namely (6,1), so the sum of the parts is 6 + 1 = 7.
%e A296966 a(8) = 16; there are two partitions of 8 into 2 distinct parts such that the smaller divides the larger, namely (7,1) and (6,2). The sum of the parts is then 7 + 1 + 6 + 2 = 16.
%e A296966 (End)
%t A296966 Table[n*Sum[(Floor[n/i] - Floor[(n - 1)/i]), {i, Floor[(n - 1)/2]}], {n, 100}]
%t A296966 f[n_] := n*Length[Select[Divisors@n, 2 # < n &]]; Array[f, 61] (* or *)
%t A296966 f[n_] := Block[{t = DivisorSigma[0, n]}, n*If[OddQ@ n, t -1, t -2]]; Array[f, 61] (* _Robert G. Wilson v_, Dec 24 2017 *)
%o A296966 (PARI) a(n) = n*sum(i=1, floor((n-1)/2), floor(n/i) - floor((n-1)/i)) \\ _Iain Fox_, Dec 22 2017
%Y A296966 Cf. A023645.
%K A296966 nonn,easy
%O A296966 1,3
%A A296966 _Wesley Ivan Hurt_, Dec 22 2017