A333697 - OEIS

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

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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