A053287 - OEIS

%I #72 Feb 17 2022 01:06:21

%S 1,2,6,8,30,36,126,128,432,600,1936,1728,8190,10584,27000,32768,

%T 131070,139968,524286,480000,1778112,2640704,8210080,6635520,32400000,

%U 44717400,113467392,132765696,533826432,534600000,2147483646,2147483648,6963536448,11452896600

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

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

%C 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

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

%H Amiram Eldar, <a href="/A053287/b053287.txt">Table of n, a(n) for n = 1..1206</a> (terms 1..100 from T. D. Noe, terms 101..250 from Jianing Song, terms 251..400 from Michel Marcus)

%H Florian Luca, <a href="http://dml.cz/dmlcz/129330">On the sum of divisors of the Mersenne numbers</a>, Mathematica Slovaca, Vol. 53. No. 5 (2003), pp. 457-466.

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

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

%F a(n) = A000010(2^n-1) or also a(n) = A062401(2^(n-1)) = phi(sigma(2^(n-1))). - _Labos Elemer_, Jul 19 2004

%p a := n -> numtheory:-phi(2^n - 1): seq(a(n), n=1..32); # _Zerinvary Lajos_, Oct 05 2007

%t EulerPhi[2^Range[25] - 1] (* _Giovanni Resta_, Sep 06 2019 *)

%o (PARI) a(n) = eulerphi(2^n-1) \\ _Michael B. Porter_, Oct 06 2009

%o (Magma) [EulerPhi(2^n-1): n in [1..40]]; // _Vincenzo Librandi_, Jul 15 2015

%o (GAP) List([1..35],n->Phi(2^n-1)); # _Muniru A Asiru_, Oct 31 2018

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

%K nonn,easy,nice

%O 1,2

%A _Labos Elemer_, Mar 03 2000