A053287 Euler totient function (A000010) of 2^n - 1.

1, 2, 6, 8, 30, 36, 126, 128, 432, 600, 1936, 1728, 8190, 10584, 27000, 32768, 131070, 139968, 524286, 480000, 1778112, 2640704, 8210080, 6635520, 32400000, 44717400, 113467392, 132765696, 533826432, 534600000, 2147483646, 2147483648, 6963536448, 11452896600

Offset: 1

Author

Labos Elemer, Mar 03 2000

Keywords

• nonn
• easy
• nice

Comments

Number of elements of multiplicative order 2^n - 1 in GF(2^n).

n divides a(n) because 2^a(n) mod 2^n - 1 is 1, 2^n mod 2^n - 1 is 1, so n | a(n). A011260(n) = a(n)/n. - Jinyuan Wang, Oct 31 2018

The set {a(n)/(2^n-1)} is dense in [0, 1] (Luca, 2003). - Amiram Eldar, Mar 04 2021

Links

• Amiram Eldar, Table of n, a(n) for n = 1..1206 (terms 1..100 from T. D. Noe, terms 101..250 from Jianing Song, terms 251..400 from Michel Marcus)
• Florian Luca, On the sum of divisors of the Mersenne numbers, Mathematica Slovaca, Vol. 53. No. 5 (2003), pp. 457-466.

Crossrefs

Cf. A000010, A000225, A051953, A057764, A011260.

Programs

GAP

List([1..35],n->Phi(2^n-1)); # Muniru A Asiru, Oct 31 2018

Magma

[EulerPhi(2^n-1): n in [1..40]]; // Vincenzo Librandi, Jul 15 2015

Maple

a := n -> numtheory:-phi(2^n - 1): seq(a(n), n=1..32); # Zerinvary Lajos, Oct 05 2007

Mathematica

EulerPhi[2^Range[25] - 1] (* Giovanni Resta, Sep 06 2019 *)

PARI

a(n) = eulerphi(2^n-1) \\ Michael B. Porter, Oct 06 2009

Formula

a(n) = A000010(A000225(n)).

a(A000079(n-1)) = A058891(n).

a(n) = A000010(2^n-1) or also a(n) = A062401(2^(n-1)) = phi(sigma(2^(n-1))). - Labos Elemer, Jul 19 2004