A214606 a(n) = gcd(n, 2^n - 2).

1, 2, 3, 2, 5, 2, 7, 2, 3, 2, 11, 2, 13, 2, 3, 2, 17, 2, 19, 2, 3, 2, 23, 2, 5, 2, 3, 14, 29, 2, 31, 2, 3, 2, 1, 2, 37, 2, 3, 2, 41, 2, 43, 2, 15, 2, 47, 2, 7, 2, 3, 2, 53, 2, 1, 2, 3, 2, 59, 2, 61, 2, 3, 2, 5, 2, 67, 2, 3, 14, 71, 2, 73, 2, 3, 2, 1, 2, 79

Offset: 1

Author

Alex Ratushnyak, Jul 22 2012

Keywords

• nonn
• easy

Comments

Greatest common divisor of n and 2^n - 2.

a(n)=n iff n=1 or n is prime or n is Fermat pseudoprime to base 2 or even pseudoprime to base 2. - Corrected by Thomas Ordowski, Jan 25 2016

Indices of 1's: A121707 preceded by 1. - False, see A267999.

Numbers n such that a(n) does not equal A020639(n) (the least prime factor of n): A146077.

Examples

			a(3) = 3 because 2^3 - 2 = 6 and gcd(3, 6) = 3.
a(4) = 2 because 2^4 - 2 = 14 and gcd(4, 14) = 2.
		

Links

• Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A000918, A064535, A121707, A020639, A146077.

Programs

Java

import java.math.BigInteger;
public class A214606 {
  public static void main (String[] args) {
    BigInteger c1 = BigInteger.valueOf(1);
    BigInteger c2 = BigInteger.valueOf(2);
    for (int n=0; n<222; n++) {
      BigInteger bn=BigInteger.valueOf(n),pm2=c1.shiftLeft(n).subtract(c2);
      System.out.printf("%s, ", bn.gcd(pm2).toString());
    }
  }
}

Magma

[GCD(n, 2^n-2): n in [1..80]]; // Vincenzo Librandi, Jan 26 2016

Maple

seq(igcd(n, (2&^n - 2) mod n), n=1 .. 1000); # Robert Israel, Jan 26 2016

Mathematica

Table[GCD[n, 2^n - 2], {n, 1, 59}] (* Alonso del Arte, Jul 22 2012 *)

PARI

a(n)=gcd(n,lift(Mod(2,n)^n-2)) \\ Charles R Greathouse IV, May 29 2014