A225679 Numerators of phi(k)/k, as k runs through the squarefree numbers (A005117).

1, 1, 2, 4, 1, 6, 2, 10, 12, 3, 8, 16, 18, 4, 5, 22, 6, 28, 4, 30, 20, 8, 24, 36, 9, 8, 40, 2, 42, 11, 46, 32, 52, 8, 12, 14, 58, 60, 15, 48, 10, 66, 44, 12, 70, 72, 18, 60, 4, 78, 20, 82, 64, 21, 56, 88, 72, 20, 23, 72, 96, 100, 16, 102, 16, 26, 106, 108, 4

Offset: 1

Author

Franz Vrabec, May 12 2013

Keywords

• nonn
• frac
• easy

Comments

To every fraction taken by the arithmetical function m -> phi(m)/m there is exactly one n such that a(n)/A225680(n) is equal to it.

Examples

			A005117(5) = 6, phi(6)/6 = 2/6 = 1/3, so a(5) = 1.
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A000010, A005117, A065463, A225680 (denominators).

Programs

Mathematica

s = Select[Range[200], SquareFreeQ]; Numerator[EulerPhi[s]/s] (* T. D. Noe, May 13 2013 *)

PARI

lista(nn) = apply(x->(numerator(eulerphi(x)/x)), Vec(select(issquarefree, [1..nn], 1))); \\ Michel Marcus, Feb 22 2021

Formula

a(n) = A000010(A005117(n))/gcd(A000010(A005117(n)),A005117(n)).

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k)/A225680(k) = Product_{p prime} (1 - 1/(p*(p+1))) = 0.7044422... (A065463). - Amiram Eldar, Nov 21 2022