%I #58 Sep 15 2024 15:59:19
%S 1,2,6,7,13,17,25,26,39,45,57,61,75,83,107,108,126,139,159,165,197,
%T 209,233,237,268,282,322,330,360,384,416,417,465,483,531,544,582,602,
%U 658,664,706,738,782,794,872,896,944,948,1005,1036,1108,1122,1176,1216
%N Sum of all odd divisors of all positive integers <= n.
%C The subsequence of primes begins: 2, 7, 13, 17, 61, 83, 107, 139, 197, 233, then no more through a(54). [_Jonathan Vos Post_, Feb 14 2010]
%C a(n) is also the total number of parts in all partitions of all positive integers <= n into an odd number of equal parts. - _Omar E. Pol_, Jun 04 2017
%H Alois P. Heinz, <a href="/A078471/b078471.txt">Table of n, a(n) for n = 1..10000</a>
%H M. Baake and R. V. Moody, <a href="https://arxiv.org/abs/math/9904028">Similarity submodules and root systems in four dimensions</a>, arXiv:math/9904028 [math.MG], 1999.
%H M. Baake and R. V. Moody, <a href="http://dx.doi.org/10.4153/CJM-1999-057-0">Similarity submodules and root systems in four dimensions</a>, Canad. J. Math. 51 (1999), 1258-1276.
%F a(n) = Sum_{k=1..n} A000593(k).
%F a(n) is asymptotic to c*n^2 where c = Pi^2/24.
%F a(n) = A024916(n) - A271342(n). - _Omar E. Pol_, Apr 08 2016
%F G.f.: (1/(1 - x))*Sum_{k>=1} k*x^k/(1 + x^k). - _Ilya Gutkovskiy_, Dec 23 2016
%F From _Ridouane Oudra_, Aug 28 2019: (Start)
%F a(n) = Sum_{k=1..n} (sigma(2k) - 2*sigma(k)), where sigma = A000203.
%F a(n) = A326124(n) - 2*A024916(n). (End)
%p with(numtheory):
%p b:= n-> add(d, d=select(x-> x::odd, divisors(n))):
%p a:= proc(n) option remember; b(n)+`if`(n=1, 0, a(n-1)) end:
%p seq(a(n), n=1..60); # _Alois P. Heinz_, Sep 25 2015
%t a[n_] := Sum[DivisorSum[k, (-1)^(# + 1) k/# &], {k, 1, n}]; Array[a, 60] (* _Jean-François Alcover_, Dec 07 2015 *)
%t Accumulate[Table[Total[Select[Divisors[n],OddQ]],{n,60}]] (* _Harvey P. Dale_, Sep 15 2024 *)
%o (PARI) a(n)=sum(v=1,n,sumdiv(v,d,(-1)^(d+1)*v/d))
%o (PARI) a(n) = sum(k=1, n, sumdiv(k, d, (d%2)*d)); \\ _Michel Marcus_, Apr 09 2016
%o (Magma) [&+[&+[d:d in Divisors(k)|IsOdd(d)]:k in [1..n]]:n in [1..60]]; // _Marius A. Burtea_, Aug 28 2019
%o (Python)
%o def A078471(n): return sum(k*(n//k) for k in range((n>>1)+1, n+1)) + sum(k*(n//k-((n>>1)//k<<1)) for k in range(1, (n>>1)+1)) # _Chai Wah Wu_, Apr 26 2023
%o (Python)
%o from math import isqrt
%o def A078471(n): return (t:=isqrt(m:=n>>1))**2*(t+1) - sum((q:=m//k)*((k<<1)+q+1) for k in range(1,t+1))-((s:=isqrt(n))**2*(s+1) - sum((q:=n//k)*((k<<1)+q+1) for k in range(1,s+1))>>1) # _Chai Wah Wu_, Oct 21 2023
%Y Partial sums of A000593.
%Y Cf. A024916, A271342, A000203, A326124.
%K nonn
%O 1,2
%A _Benoit Cloitre_, Dec 31 2002
%E Better definition from _Omar E. Pol_, Apr 09 2016