A333697 a(n) = Sum_{d|n} (-1)^omega(n/d) * phi(rad(n/d)) * p(d), where p = A000041 (partition numbers).

1, 1, 1, 2, 3, 6, 9, 14, 22, 31, 46, 59, 89, 114, 158, 201, 281, 337, 472, 570, 756, 936, 1233, 1456, 1926, 2323, 2942, 3556, 4537, 5334, 6812, 8088, 10021, 11997, 14805, 17432, 21601, 25507, 30971, 36606, 44543, 52106, 63219, 74097, 88680, 104281, 124708, 145205, 173429, 202124

Offset

1,4

Links

Table of n, a(n) for n=1..50.

Formula

a(n) = Sum_{d|n} A023900(n/d) * A000041(d).

a(n) = Sum_{d|n} A047968(n/d) * mu(d) * d.

Sum_{k=1..n} a(gcd(n,k)) = A000041(n).

Mathematica

Table[Sum[(-1)^PrimeNu[n/d] EulerPhi[Last[Select[Divisors[n/d], SquareFreeQ]]] PartitionsP[d], {d, Divisors[n]}], {n, 50}]

Prog

(PARI) rad(n) = factorback(factorint(n)[, 1]); \\ A007947
a(n) = sumdiv(n, d, (-1)^omega(n/d) * eulerphi(rad(n/d)) * numbpart(d)); \\ Michel Marcus, Apr 03 2020

Crossrefs

Cf. A000010, A000041, A001221, A007947, A008683, A023900, A047968, A078392.

Sequence in context: A173303 A308251 A058609 * A128518 A022567 A134004

Adjacent sequences: A333694 A333695 A333696 * A333698 A333699 A333700

Keyword

nonn

Author

Ilya Gutkovskiy, Apr 02 2020

Status

approved