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

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, 42, 31, 24, 30, 72, 32, 43, 48, 54, 48, 12, 38, 60, 56, 66, 42, 96, 44, 36, 24, 72, 48, 44, 8, 18, 72, 42, 54, 93, 72, 88, 80, 90, 60, 72, 62, 96, 32, 43, 84, 144, 68, 54, 96, 144

Offset: 1

Author

Yasutoshi Kohmoto

Keywords

• nonn
• mult

Comments

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

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

Examples

			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.
		

Links

• Antti Karttunen, Table of n, a(n) for n = 1..16384
• Index entries for sequences related to sums of divisors.

Crossrefs

Cf. A000203, A008864, A072587, A181150, A268335, A295316.

Cf. also A126849.

Cf. A065463, A072691.

Programs

Maple

A033634 := proc(n)
    a := 1 ;
    for d in ifactors(n)[2] do
        if type(op(2,d),'odd') then
            s := op(2,d) ;
        else
            s := op(2,d)-1 ;
         end if;
        p := op(1,d) ;
        a := a*(1+(p^(s+2)-p)/(p^2-1)) ;
    end do:
    a;
end proc: # R. J. Mathar, Nov 20 2010

Mathematica

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 *)

PARI

A295316(n) = factorback(apply(e -> (e%2), factorint(n)[, 2]));
A033634(n) = sumdiv(n,d,A295316(d)*d); \\ Antti Karttunen, Nov 23 2017

Formula

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:

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

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

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

a(n) = Sum_{d|n} A295316(d)*d. - Antti Karttunen, Nov 23 2017

a(n) <= A000203(n). - R. J. Mathar, Nov 27 2017

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