|
|
A323599
|
|
Dirichlet convolution of the identity function with omega.
|
|
15
|
|
|
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
|
|
|
FORMULA
|
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)):
|
|
MATHEMATICA
|
|
|
PROG
|
(PARI) a(n) = sumdiv(n, d, d*omega(n/d)); \\ Michel Marcus, Jan 22 2019
|
|
CROSSREFS
|
Inverse Möbius transform of A069359.
|
|
KEYWORD
|
nonn,changed
|
|
AUTHOR
|
|
|
STATUS
|
approved
|
|
|
|