A345315 a(n) = Sum_{d|n} d^[Omega(d) = 2], where [ ] is the Iverson bracket.

1, 2, 2, 6, 2, 9, 2, 7, 11, 13, 2, 14, 2, 17, 18, 8, 2, 19, 2, 18, 24, 25, 2, 16, 27, 29, 12, 22, 2, 36, 2, 9, 36, 37, 38, 25, 2, 41, 42, 20, 2, 46, 2, 30, 28, 49, 2, 18, 51, 39, 54, 34, 2, 21, 58, 24, 60, 61, 2, 43, 2, 65, 34, 10, 68, 66, 2, 42, 72, 64, 2, 28, 2, 77, 44, 46, 80

Offset

1,2

Comments

For each divisor d of n, add d if d is semiprime, otherwise add 1. For example, the divisors of 24 are: 1, 2, 3, 4, 6, 8, 12, 24, and the only semiprime divisors of 24 are 4 and 6, so a(24) = 1 + 1 + 1 + 4 + 6 + 1 + 1 + 1 = 16.

Links

Table of n, a(n) for n=1..77.

Formula

a(p) = Sum_{d|p} d^[Omega(d) = 2] = 1^0 + p^0 = 2, for primes p.

Example

a(n) = Sum_{d|12} d^[Omega(d) = 2] = 1^0 + 2^0 + 3^0 + 4^1 + 6^1 + 12^0 = 14.

Mathematica

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

Prog

(PARI) a(n) = sumdiv(n, d, if (bigomega(d)==2, d, 1)); \\ Michel Marcus, Jun 13 2021

Crossrefs

Cf. A001222 (Omega).

Sequence in context: A291840 A208448 A096869 * A154009 A297792 A266722

Adjacent sequences: A345312 A345313 A345314 * A345316 A345317 A345318

Keyword

nonn

Author

Wesley Ivan Hurt, Jun 13 2021

Status

approved