A345345 a(n) = Sum_{d^2|n} omega(n/d^2).

0, 1, 1, 1, 1, 2, 1, 2, 1, 2, 1, 3, 1, 2, 2, 2, 1, 3, 1, 3, 2, 2, 1, 4, 1, 2, 2, 3, 1, 3, 1, 3, 2, 2, 2, 4, 1, 2, 2, 4, 1, 3, 1, 3, 3, 2, 1, 5, 1, 3, 2, 3, 1, 4, 2, 4, 2, 2, 1, 5, 1, 2, 3, 3, 2, 3, 1, 3, 2, 3, 1, 6, 1, 2, 3, 3, 2, 3, 1, 5, 2, 2, 1, 5, 2, 2, 2, 4, 1, 5, 2, 3

Offset

1,6

Links

Table of n, a(n) for n=1..92.

Formula

a(p) = Sum_{d^2|p} omega(p/d^2) = omega(p) = 1 for p prime.

Sum_{k=1..n} a(k) = Sum_{k=1..n} omega(k) * floor(sqrt(n/k)) = Sum_{k=1..floor(sqrt(n))} A013939(floor(n / k^2)). - Daniel Suteu, May 11 2026

Example

a(12) = Sum_{d^2|12} omega(12/d^2) = omega(12) + omega(3) = 2 + 1 = 3.

Mathematica

Table[Sum[PrimeNu[n/k^2] (1 - Ceiling[n/k^2] + Floor[n/k^2]), {k, n}], {n, 100}]

Prog

(PARI) a(n) = sumdiv(n, d, if (issquare(d), omega(n/d))); \\ Michel Marcus, Jun 14 2021

Crossrefs

Cf. A001221 (omega), A062799, A345344.

Sequence in context: A341596 A099042 A140774 * A056924 A342083 A381452

Adjacent sequences: A345342 A345343 A345344 * A345346 A345347 A345348

Keyword

nonn

Author

Wesley Ivan Hurt, Jun 14 2021

Status

approved