A323599 Dirichlet convolution of the identity function with omega.

0, 1, 1, 3, 1, 7, 1, 7, 4, 9, 1, 19, 1, 11, 10, 15, 1, 25, 1, 25, 12, 15, 1, 43, 6, 17, 13, 31, 1, 54, 1, 31, 16, 21, 14, 67, 1, 23, 18, 57, 1, 68, 1, 43, 37, 27, 1, 91, 8, 49, 22, 49, 1, 79, 18, 71, 24, 33, 1, 142, 1, 35, 45, 63, 20, 96, 1, 61, 28, 90, 1, 151, 1, 41, 55

Offset: 1

Author

Torlach Rush, Jan 18 2019

Keywords

• nonn

Comments

a(n) = omega(n) = 1 iff n is prime.

a(n) = A323600(n) = 1 iff n is prime.

a(n) = A323600(n) - 1 = 1 iff n is the square of a prime.

a(n) = A323600(n) - 2 = 2 iff n is a squarefree semiprime.

a(n) = A323600(n) - (p + 2) if n is the cube of a prime p.

Links

• Antti Karttunen, Table of n, a(n) for n = 1..16384
• Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537

Crossrefs

Cf. A001221, A180253, A276085, A319684, A323600, A329347, A329380.

Inverse Möbius transform of A069359.

Programs

Maple

with(numtheory):
a:= n-> add(d*nops(factorset(n/d)), d=divisors(n)):
seq(a(n), n=1..100);  # Alois P. Heinz, Jan 28 2019

Mathematica

Table[DivisorSum[n, # PrimeNu[n/#] &], {n, 75}] (* Michael De Vlieger, Jan 27 2019 *)

PARI

a(n) = sumdiv(n, d, d*omega(n/d)); \\ Michel Marcus, Jan 22 2019

Formula

a(n) = Sum_{d|n} d * A001221(n/d).

a(n) = Sum_{p|n} sigma(n/p) where p is prime and sigma(n) = A000203(n). - Ridouane Oudra, Apr 28 2019

a(n) = Sum_{d|n} A069359(d), a(n) = A276085(A329380(n)). - Antti Karttunen, Nov 12 2019

From Torlach Rush, Mar 23 2024: (Start)

For p in primes: (Start)

a(p^(k+1)) = a(p^k) + p^k, k >= 0.

a(p^2) = p + 1.

(End)

a(2^k) = 2^k - 1, k >= 0.

(End)