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A323599 Dirichlet convolution of the identity function with omega. 9
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 (list; graph; refs; listen; history; text; internal format)
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

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: A324865 A316553 A186428 * A167515 A140435 A194181

Adjacent sequences:  A323596 A323597 A323598 * A323600 A323601 A323602

KEYWORD

nonn

AUTHOR

Torlach Rush, Jan 18 2019

STATUS

approved

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Last modified August 6 11:55 EDT 2020. Contains 336246 sequences. (Running on oeis4.)