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A033634 OddPowerSigma(n) = sum of odd power divisors of n.

%I A033634 #33 Oct 27 2022 07:31:52
%S A033634 1,3,4,3,6,12,8,11,4,18,12,12,14,24,24,11,18,12,20,18,32,36,24,44,6,
%T A033634 42,31,24,30,72,32,43,48,54,48,12,38,60,56,66,42,96,44,36,24,72,48,44,
%U A033634 8,18,72,42,54,93,72,88,80,90,60,72,62,96,32,43,84,144,68,54,96,144
%N A033634 OddPowerSigma(n) = sum of odd power divisors of n.
%C A033634 Odd power divisors of n are all the terms of A268335 (numbers whose prime power factorization contains only odd exponents) that divide n. - _Antti Karttunen_, Nov 23 2017
%C A033634 The Mobius transform is 1, 2, 3, 0, 5, 6, 7, 8, 0, 10, 11, 0, 13, 14, 15, 0, 17, 0, 19, 0, 21, 22, 23, 24, 0, 26, ..., where the places of zeros seem to be listed in A072587. - _R. J. Mathar_, Nov 27 2017
%H A033634 Antti Karttunen, <a href="/A033634/b033634.txt">Table of n, a(n) for n = 1..16384</a>
%H A033634 <a href="/index/Su#sums_of_divisors">Index entries for sequences related to sums of divisors</a>.
%F A033634 Let n = Product p(i)^r(i) then a(n) = Product (1+[p(i)^(s(i)+2)-p(i)]/[p(i)^2-1]), where si=ri when ri is odd, si=ri-1 when ri is even. Special cases:
%F A033634 a(p) = 1+p for primes p, subsequence A008864.
%F A033634 a(p^2) = 1+p for primes p.
%F A033634 a(p^3) = 1+p+p^3 for primes p, subsequence A181150.
%F A033634 a(n) = Sum_{d|n} A295316(d)*d. - _Antti Karttunen_, Nov 23 2017
%F A033634 a(n) <= A000203(n). - _R. J. Mathar_, Nov 27 2017
%F A033634 Sum_{k=1..n} a(k) ~ c * n^2, where c = (Pi^2/12) * Product_{p prime} (1 - 1/(p*(p+1))) = A072691 * A065463 = 0.5793804... . - _Amiram Eldar_, Oct 27 2022
%e A033634 The divisors of 7 are 1^1 and 7^1, which have only odd exponents (=1), so a(8) = 1+7 = 8. The divisors of 8 are 1^1, 2^1, 2^2 and 2^3; 2^2 has an even prime power and does not count, so a(8) = 1+2+8=11. The divisors of 12 are 1^1, 2^1, 3^1, 2^2, 2^1*3^1 and 2^2*3; 2^2 and 2^2*3 don't count because they have prime factors with even powers, so a(12) = 1+2+3+6 = 12.
%p A033634 A033634 := proc(n)
%p A033634     a := 1 ;
%p A033634     for d in ifactors(n)[2] do
%p A033634         if type(op(2,d),'odd') then
%p A033634             s := op(2,d) ;
%p A033634         else
%p A033634             s := op(2,d)-1 ;
%p A033634          end if;
%p A033634         p := op(1,d) ;
%p A033634         a := a*(1+(p^(s+2)-p)/(p^2-1)) ;
%p A033634     end do:
%p A033634     a;
%p A033634 end proc: # _R. J. Mathar_, Nov 20 2010
%t A033634 f[e_] := If[OddQ[e], e+2, e+1]; fun[p_,e_] := 1 + (p^f[e] - p)/(p^2 - 1); a[1] = 1; a[n_] := Times @@ (fun @@@ FactorInteger[n]); Array[a, 100] (* _Amiram Eldar_, May 14 2019 *)
%o A033634 (PARI)
%o A033634 A295316(n) = factorback(apply(e -> (e%2), factorint(n)[, 2]));
%o A033634 A033634(n) = sumdiv(n,d,A295316(d)*d); \\ _Antti Karttunen_, Nov 23 2017
%Y A033634 Cf. A000203, A008864, A072587, A181150, A268335, A295316.
%Y A033634 Cf. also A126849.
%Y A033634 Cf. A065463, A072691.
%K A033634 nonn,mult
%O A033634 1,2
%A A033634 _Yasutoshi Kohmoto_