A057764 -id:A057764 - cp's OEIS frontend

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

Labos Elemer, Mar 03 2000

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

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.

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

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

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

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

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

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

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

A136673 Triangle of coefficients from a polynomial recursion for Galois field GF(2^n) polynomials: p(x,n)=(x+1)p(x,n-1)-xp(x,n-2); or f(x,n)=x^n+x+1;.

Original entry on oeis.org

2, 1, 1, 2, 1, 1, 1, 1, 1, 0, 1, 1, 1, 0, 0, 1, 1, 1, 0, 0, 0, 1, 1, 1, 0, 0, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1

Offset: 1

Roger L. Bagula, Apr 05 2008

Row sums are:
{3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3}
The result is very dependent on the two initial polynomials.

			{2, 1},
{1, 2},
{1, 1, 1},
{1, 1, 0, 1},
{1, 1, 0, 0, 1},
{1, 1, 0, 0, 0, 1},
{1, 1, 0, 0, 0, 0, 1},
{1, 1, 0, 0, 0, 0, 0, 1},
{1, 1, 0, 0, 0, 0, 0, 0, 1},
{1, 1, 0, 0, 0, 0, 0, 0, 0, 1},
{1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1}

Taylor L. Booth, Sequential Machines and Automata Theory, John Wiley and Sons, Inc., 1967, Appendix I

Cf. A057764, A103204.

p[x, 0] = 2 + x; p[x, 1] = 1 + 2*x; p[x_, n_] := p[x, n] = (x + 1)*p[x, n - 1] - x*p[x, n - 2]; Table[ExpandAll[p[x, n]], {n, 0, 10}]; a = Table[CoefficientList[p[x, n], x], {n, 0, 10}] Flatten[a]

p(x,0)=2+x;p(x,1)=1+2*x; p(x,n)=(x+1)*p(x,n-1)-x*p(x,n-2); or f(x,n)=x^n+x+1;

cp's OEIS Frontend

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

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

GAP

Magma

Maple

Mathematica

PARI

Formula

A136673 Triangle of coefficients from a polynomial recursion for Galois field GF(2^n) polynomials: p(x,n)=(x+1)p(x,n-1)-xp(x,n-2); or f(x,n)=x^n+x+1;.

Views

Author

Keywords

Comments

Examples

References

Crossrefs

Programs

Mathematica

Formula

cp's OEIS Frontend

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

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

GAP

Magma

Maple

Mathematica

PARI

Formula

A136673 Triangle of coefficients from a polynomial recursion for Galois field GF(2^n) polynomials: p(x,n)=(x+1)*p(x,n-1)-x*p(x,n-2); or f(x,n)=x^n+x+1;.

Views

Author

Keywords

Comments

Examples

References

Crossrefs

Programs

Mathematica

Formula

A136673 Triangle of coefficients from a polynomial recursion for Galois field GF(2^n) polynomials: p(x,n)=(x+1)p(x,n-1)-xp(x,n-2); or f(x,n)=x^n+x+1;.