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

1,4

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

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)

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 *)

Prog

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

Crossrefs

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

Inverse Möbius transform of A069359.

Sequence in context: A347959 A347963 A186428 * A167515 A140435 A194181

Adjacent sequences: A323596 A323597 A323598 * A323600 A323601 A323602

Keyword

nonn

Author

Torlach Rush, Jan 18 2019

Status

approved