A345220 Number of divisors d of n with an even number of primes not exceeding d.

1, 1, 2, 2, 1, 2, 2, 3, 3, 2, 1, 3, 2, 3, 3, 4, 1, 3, 2, 4, 4, 2, 1, 4, 1, 2, 3, 4, 2, 5, 1, 4, 2, 1, 2, 4, 2, 3, 4, 6, 1, 5, 2, 4, 5, 2, 1, 5, 2, 2, 2, 3, 2, 4, 2, 6, 4, 3, 1, 7, 2, 2, 6, 5, 3, 4, 1, 2, 2, 4, 2, 6, 1, 2, 3, 4, 2, 4, 2, 8, 4, 2, 1, 6, 1, 2, 3, 5, 2, 8, 4, 4, 3

Offset

1,3

Comments

Inverse Möbius transform of (pi(n)+1) mod 2 = A131377(n). - Wesley Ivan Hurt, Jul 04 2025

Links

Table of n, a(n) for n=1..93.

Formula

a(n) = Sum_{d|n} ((pi(d)+1) mod 2).

a(n) = A000005(n) - A345219(n). - Wesley Ivan Hurt, Jul 05 2025

Example

a(24) = 4; The divisors d of 24 are: 1, 2, 3, 4, 6, 8, 12, 24 and the corresponding values of pi(d) are: 0, 1, 2, 2, 3, 4, 5, 9. There are 4 even values of pi(d).

Mathematica

Table[Sum[Mod[PrimePi[d] + 1, 2], {d, Divisors[n]}], {n, 100}]

Prog

(PARI) a(n) = sumdiv(n, d, !(primepi(d) % 2)); \\ Michel Marcus, Jun 11 2021

Crossrefs

Cf. A000005 (tau), A000720 (pi), A131377, A345219.

Sequence in context: A245574 A245573 A236968 * A265744 A331083 A245588

Adjacent sequences: A345217 A345218 A345219 * A345221 A345222 A345223

Keyword

nonn

Author

Wesley Ivan Hurt, Jun 11 2021

Status

approved