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.
%I A296955 #48 Nov 24 2024 00:04:54
%S A296955 0,0,1,1,1,3,1,3,4,3,1,10,1,3,9,7,1,12,1,12,11,3,1,24,6,3,13,14,1,27,
%T A296955 1,15,15,3,13,37,1,3,17,30,1,33,1,18,33,3,1,52,8,18,21,20,1,39,17,36,
%U A296955 23,3,1,78,1,3,41,31,19,45,1,24,27,39,1,87,1,3,49,26,19,51,1,66,40,3,1,98,23,3,33,48,1,99,21,30,35,3,25,108,1
%N A296955 Sum of the smaller parts of the partitions of n into two distinct parts such that the smaller part divides the larger.
%C A296955 The number of partitions of n into 3 parts whose "middle" part divides n. - _Wesley Ivan Hurt_, Oct 21 2021
%H A296955 Antti Karttunen, <a href="/A296955/b296955.txt">Table of n, a(n) for n = 1..65537</a>
%H A296955 Joerg Arndt, <a href="http://arxiv.org/abs/1202.6525">On computing the generalized Lambert series</a>, arXiv:1202.6525v3 [math.CA], (2012).
%H A296955 <a href="/index/Par#part">Index entries for sequences related to partitions</a>
%F A296955 a(n) = Sum_{i=1..floor((n-1)/2)} i * (floor(n/i) - floor((n-1)/i)).
%F A296955 a(n) = the sum of the divisors < n/2. - _Robert G. Wilson v_, Dec 23 2017
%F A296955 a(n) = 1 iff n is an odd prime or n=4. - _Robert G. Wilson v_, Dec 23 2017
%F A296955 G.f.: Sum_{k>=1} k * x^(3*k) / (1 - x^k). - _Ilya Gutkovskiy_, May 30 2020
%F A296955 G.f.: Sum_{k >= 3} x^k/(1 - x^k)^2. Cf. A023645. - _Peter Bala_, Jan 13 2021
%F A296955 Faster converging g.f.: Sum_{n >= 1} q^(n*(n+2))*( n*q^(3*n+4) - (n + 1)*q^(2*n+2) - (n - 1)*q^(n+2) + n )/( (1 - q^n )*(1 - q^(n+2))^2 ). (In equation 1 in Arndt, after combining the two n = 0 summands to get t/(1 - t), apply the operator t*d/dt and then set t = q^2 and x = 1. Cf. A001065.) - _Peter Bala_, Jan 22 2021
%F A296955 a(n) = A000203(n) - A080512(n). - _Ridouane Oudra_, Aug 15 2024
%e A296955 a(12) = 10; the partitions of 12 into two distinct parts are (11,1), (10,2), (9,3), (8,4) and (7,5). 1 divides 11, 2 divides 10, 3 divides 9 and 4 divides 8, so the sum of the smaller parts gives 1 + 2 + 3 + 4 = 10.
%p A296955 with(numtheory):
%p A296955 a := n -> add( d, d = divisors(n) minus {floor((n+1)/2), n} ):
%p A296955 seq(a(n), n = 1..100); # _Peter Bala_, Jan 13 2021
%t A296955 Table[Sum[i (Floor[n/i] - Floor[(n - 1)/i]), {i, Floor[(n - 1)/2]}], {n, 100}]
%t A296955 f[n_] := Plus @@ Select[Divisors@n, 2 # < n &]; Array[f, 75] (* _Robert G. Wilson v_, Dec 23 2017 *)
%o A296955 (PARI) A296955(n) = sumdiv(n,d,(d<(n/2))*d); \\ _Antti Karttunen_, Sep 25 2018
%Y A296955 Cf. A297024, A001065, A023645, A000203, A080512.
%K A296955 nonn,easy
%O A296955 1,6
%A A296955 _Wesley Ivan Hurt_, Dec 22 2017
%E A296955 More terms from _Antti Karttunen_, Sep 25 2018