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

1,2

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

Links

Antti Karttunen, Table of n, a(n) for n = 1..16384

Index entries for sequences related to sums of divisors.

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

Example

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.

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 *)

Prog

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

Crossrefs

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

Cf. also A126849.

Cf. A065463, A072691.

Sequence in context: A000113 A365347 A069915 * A366764 A365337 A349337

Adjacent sequences: A033631 A033632 A033633 * A033635 A033636 A033637

Keyword

nonn,mult

Author

Yasutoshi Kohmoto

Status

approved