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 A033634 OddPowerSigma(n) = sum of odd power divisors of n. 3

%I

%S 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 42,31,24,30,72,32,43,48,54,48,12,38,60,56,66,42,96,44,36,24,72,48,44,

%U 8,18,72,42,54,93,72,88,80,90,60,72,62,96,32,43,84,144,68,54,96,144

%N OddPowerSigma(n) = sum of odd power divisors of n.

%C 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 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 Antti Karttunen, <a href="/A033634/b033634.txt">Table of n, a(n) for n = 1..16384</a>

%H <a href="/index/Su#sums_of_divisors">Index entries for sequences related to sums of divisors</a>

%F 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 a(p) = 1+p for primes p, subsequence A008864.

%F a(p^2) = 1+p for primes p.

%F a(p^3) = 1+p+p^3 for primes p, subsequence A181150.

%F a(n) = Sum_{d|n} A295316(d)*d. - _Antti Karttunen_, Nov 23 2017

%F a(n) <= A000203(n). - _R. J. Mathar_, Nov 27 2017

%e 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 := proc(n)

%p a := 1 ;

%p for d in ifactors(n)[2] do

%p if type(op(2,d),'odd') then

%p s := op(2,d) ;

%p else

%p s := op(2,d)-1 ;

%p end if;

%p p := op(1,d) ;

%p a := a*(1+(p^(s+2)-p)/(p^2-1)) ;

%p end do:

%p a;

%p end proc: # _R. J. Mathar_, Nov 20 2010

%t 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 (PARI)

%o A295316(n) = factorback(apply(e -> (e%2), factorint(n)[, 2]));

%o A033634(n) = sumdiv(n,d,A295316(d)*d); \\ _Antti Karttunen_, Nov 23 2017

%Y Cf. A268335, A295316.

%Y Cf. also A126849.

%K nonn,mult

%O 1,2

%A _Yasutoshi Kohmoto_

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Last modified July 16 04:26 EDT 2019. Contains 325064 sequences. (Running on oeis4.)