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

1,3

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.

Links

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

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

Example

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

Mathematica

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

Prog

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

Crossrefs

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

Sequence in context: A096907 A360555 A249146 * A081879 A066248 A065164

Adjacent sequences: A225676 A225677 A225678 * A225680 A225681 A225682

Keyword

nonn,frac,easy

Author

Franz Vrabec, May 12 2013

Status

approved