A053139 - OEIS

A053139

a(n) = phi(n) - mu(n).

0, 2, 3, 2, 5, 1, 7, 4, 6, 3, 11, 4, 13, 5, 7, 8, 17, 6, 19, 8, 11, 9, 23, 8, 20, 11, 18, 12, 29, 9, 31, 16, 19, 15, 23, 12, 37, 17, 23, 16, 41, 13, 43, 20, 24, 21, 47, 16, 42, 20, 31, 24, 53, 18, 39, 24, 35, 27, 59, 16, 61, 29, 36, 32, 47, 21, 67, 32

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

LINKS

G. C. Greubel, Table of n, a(n) for n = 1..10000

Robert P. Schneider, A golden product identity for e, Mathematics Magazine, Vol. 87, No. 2 (2014), pp. 132-134.

FORMULA

a(n) = n if and only if n is a prime.

Product_{n>=1} (1 - 1/phi^n)^(-a(n)/n) = e, where phi is the golden ratio (A001622) (Schneider, 2014). - Amiram Eldar, Jun 11 2022

MATHEMATICA

a[ n_] := EulerPhi @ n - MoebiusMu @ n; Table[a[n], {n, 70}] (* Michael Somos, Jul 19 2011 *)

PROG

(PARI) {a(n) = if( n==0, 0, eulerphi( n) - moebius( n))} /* Michael Somos, Jul 19 2011 */

(Magma) [EulerPhi(n) - MoebiusMu(n): n in [1..100]]; // G. C. Greubel, Sep 03 2018

CROSSREFS

Cf. A000010, A008683.

Cf. A001113, A001622.

Sequence in context: A332424 A293212 A157449 * A127705 A124386 A098668

Adjacent sequences: A053136 A053137 A053138 * A053140 A053141 A053142

KEYWORD

nonn

AUTHOR

N. J. A. Sloane, Mar 25 2000

STATUS

approved