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A352047 Sum of the divisor complements of the odd proper divisors of n.

%I A352047 #33 Oct 13 2023 06:52:53
%S A352047 0,2,3,4,5,8,7,8,12,12,11,16,13,16,23,16,17,26,19,24,31,24,23,32,30,
%T A352047 28,39,32,29,48,31,32,47,36,47,52,37,40,55,48,41,64,43,48,77,48,47,64,
%U A352047 56,62,71,56,53,80,71,64,79,60,59,96,61,64,103,64,83,96,67,72,95,96,71,104
%N A352047 Sum of the divisor complements of the odd proper divisors of n.
%H A352047 Michael De Vlieger, <a href="/A352047/b352047.txt">Table of n, a(n) for n = 1..10000</a>
%F A352047 a(n) = n * Sum_{d|n, d<n, d odd} 1 / d.
%F A352047 a(2n+1) = A000593(2n+1) - 1. - _Chai Wah Wu_, Mar 01 2022
%F A352047 G.f.: Sum_{k>=2} k * x^k / (1 - x^(2*k)). - _Ilya Gutkovskiy_, May 14 2023
%F A352047 exp( 2*Sum_{n>=1} a(n)*x^n/n ) is the g.f. of A300415. - _Paul D. Hanna_, May 15 2023
%F A352047 From _Amiram Eldar_, Oct 13 2023: (Start)
%F A352047 a(n) = A000593(n) * A006519(n) - A000035(n).
%F A352047 Sum_{k=1..n} a(k) = c * n^2 / 2, where c = Pi^2/8 = 1.23370055... (A111003). (End)
%e A352047 a(10) = 12; a(10) = 10 * Sum_{d|10, d<10, d odd} 1 / d = 10 * (1/1 + 1/5) = 12.
%p A352047 A352047 := proc(n)
%p A352047     local a,d ;
%p A352047     a := 0 ;
%p A352047     for d in numtheory[divisors](n) do
%p A352047         if type(d,'odd') and d < n then
%p A352047             a := a+n/d ;
%p A352047         end if;
%p A352047     end do:
%p A352047     a ;
%p A352047 end proc:
%p A352047 seq(A352047(n),n=1..30) ; # _R. J. Mathar_, Mar 09 2022
%t A352047 Table[n DivisorSum[n, 1/# &, # < n && OddQ[#] &], {n, 72}] (* _Michael De Vlieger_, Mar 02 2022 *)
%t A352047 a[n_] := DivisorSigma[-1, n / 2^IntegerExponent[n, 2]] * n - Mod[n, 2]; Array[a, 100] (* _Amiram Eldar_, Oct 13 2023 *)
%o A352047 (PARI) a(n) = n*sumdiv(n, d, if ((d%2) && (d<n), 1/d)); \\ _Michel Marcus_, Mar 02 2022
%o A352047 (PARI) a(n) = n * sigma(n >> valuation(n, 2), -1) - n % 2; \\ _Amiram Eldar_, Oct 13 2023
%o A352047 (Python)
%o A352047 from math import prod
%o A352047 from sympy import factorint
%o A352047 def A352047(n): return prod(p**e if p == 2 else (p**(e+1)-1)//(p-1) for p, e in factorint(n).items()) - n % 2 # _Chai Wah Wu_, Mar 02 2022
%Y A352047 Cf. A000035, A000593, A002131, A006519, A111003, A300415.
%Y A352047 Sum of the k-th powers of the divisor complements of the odd proper divisors of n for k=0..10: A091954 (k=0), this sequence (k=1), A352048 (k=2), A352049 (k=3), A352050 (k=4), A352051 (k=5), A352052 (k=6), A352053 (k=7), A352054 (k=8), A352055 (k=9), A352056 (k=10).
%K A352047 nonn,easy
%O A352047 1,2
%A A352047 _Wesley Ivan Hurt_, Mar 01 2022