A369257 a(n) = number of odd divisors of n that have an even number of prime factors with multiplicity.

1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 1, 2, 1, 1, 2, 1, 2, 2, 1, 1, 2, 1, 1, 2, 1, 1, 3, 1, 1, 1, 2, 2, 2, 1, 1, 2, 2, 1, 2, 1, 1, 2, 1, 1, 3, 1, 2, 2, 1, 1, 2, 2, 1, 2, 1, 1, 3, 1, 2, 2, 1, 1, 3, 1, 1, 2, 2, 1, 2, 1, 1, 3, 2, 1, 2, 1, 2, 1, 1, 2, 3, 2, 1, 2, 1, 1, 4

Offset

1,9

Links

Antti Karttunen, Table of n, a(n) for n = 1..65537

Formula

a(n) = Sum_{d|n} A353557(d).

a(n) = A001227(n) - A369258(n).

a(n) = a(2*n) = a(A000265(n)).

For n >= 1, a(2n-1) = A038548(2n-1); for n > 1, a(2n) < A038548(2n).

From _Antti Karttunen_, Jan 27 2024: (Start)

a(n) = A038548(A000265(n)).

a(n) = (A001227(n)+A053866(n))/2.

Dirichlet g.f.: (zeta(s)^2*(1-2^-s) + zeta(2s)*(1+2^-s)) / 2.

(End)

Example

Of the eight odd divisors of 105, the four divisors 1, 15, 21, 35 all have an even number of prime factors (A001222(d) is even), therefore a(105) = 4.

Prog

(PARI)
A353557(n) = ((n%2)&&(!(bigomega(n)%2)));
A369257(n) = sumdiv(n, d, A353557(d));

Crossrefs

Inverse Möbius transform of A353557.

Cf. A000265, A001227, A038548, A046337, A053866, A353557, A369258, A369454 (Dirichlet inverse).

Sequence in context: A300820 A356936 A257990 * A257743 A033272 A324907

Adjacent sequences: A369254 A369255 A369256 * A369258 A369259 A369260

Keyword

nonn

Author

_Antti Karttunen_, Jan 24 2024

Status

approved