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

1,2

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.

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

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 *)

Prog

(PARI) a(n) = eulerphi(2^n-1) \\ Michael B. Porter, Oct 06 2009
(Magma) [EulerPhi(2^n-1): n in [1..40]]; // Vincenzo Librandi, Jul 15 2015
(GAP) List([1..35], n->Phi(2^n-1)); # Muniru A Asiru, Oct 31 2018

Crossrefs

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

Sequence in context: A321471 A216762 A132269 * A086323 A336618 A075999

Adjacent sequences: A053284 A053285 A053286 * A053288 A053289 A053290

Keyword

nonn,easy,nice

Author

Labos Elemer, Mar 03 2000

Status

approved