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
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]));
CROSSREFS
KEYWORD
nonn,mult
AUTHOR
STATUS
approved