A333698 G.f.: Sum_{k>=1} phi(k) * x^prime(k) / (1 - x^prime(k)).

0, 1, 1, 1, 2, 2, 2, 1, 1, 3, 4, 2, 2, 3, 3, 1, 6, 2, 4, 3, 3, 5, 6, 2, 2, 3, 1, 3, 4, 4, 10, 1, 5, 7, 4, 2, 4, 5, 3, 3, 12, 4, 6, 5, 3, 7, 8, 2, 2, 3, 7, 3, 8, 2, 6, 3, 5, 5, 16, 4, 6, 11, 3, 1, 4, 6, 18, 7, 7, 5, 8, 2, 12, 5, 3, 5, 6, 4, 10, 3, 1, 13, 22, 4, 8

Offset

1,5

Links

Table of n, a(n) for n=1..85.

Index entries for sequences computed from indices in prime factorization

Formula

If n = Product (p_j^k_j) then a(n) = Sum (phi(pi(p_j))).

Example

a(63) = a(3^2 * 7) = a(prime(2)^2 * prime(4)) = A000010(2) + A000010(4) = 1 + 2 = 3.

Mathematica

nmax = 85; CoefficientList[Series[Sum[EulerPhi[k] x^Prime[k]/(1 - x^Prime[k]), {k, 1, nmax}], {x, 0, nmax}], x] // Rest
a[n_] := Plus @@ (EulerPhi[PrimePi[#[[1]]]] & /@ FactorInteger[n]); Table[a[n], {n, 85}]

Prog

(PARI) a(n) = my(f=factor(n)); sum(k=1, #f~, eulerphi(primepi(f[k, 1]))); \\ Michel Marcus, Apr 03 2020

Crossrefs

Cf. A000010, A000720, A055631, A066328, A067240.

Sequence in context: A076545 A162246 A277447 * A213126 A118400 A159853

Adjacent sequences: A333695 A333696 A333697 * A333699 A333700 A333701

Keyword

nonn

Author

Ilya Gutkovskiy, Apr 02 2020

Status

approved