A146077 -id:A146077 - cp's OEIS frontend

A159353 a(n) = the smallest positive integer such that a(n)*(2^n - 2) is a multiple of n.

1, 1, 1, 2, 1, 3, 1, 4, 3, 5, 1, 6, 1, 7, 5, 8, 1, 9, 1, 10, 7, 11, 1, 12, 5, 13, 9, 2, 1, 15, 1, 16, 11, 17, 35, 18, 1, 19, 13, 20, 1, 21, 1, 22, 3, 23, 1, 24, 7, 25, 17, 26, 1, 27, 55, 28, 19, 29, 1, 30, 1, 31, 21, 32, 13, 33, 1, 34, 23, 5, 1, 36, 1, 37, 25, 38, 77, 39, 1, 40, 27, 41, 1, 42

Offset: 1

Leroy Quet, Apr 11 2009

nonn

This is not the same as sequence A032742, where A032742(n) = the largest proper divisor of n. See A146077 for indices at which A032742 and this sequence differ.

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Cf. A000918, A146077.

[Denominator((2^n-2)/n): n in [1..84]]; // Juri-Stepan Gerasimov, Sep 09 2014

Array[Block[{k = 1}, While[! Divisible[k (2^# - 2), #], k++]; k] &, 84] (* Michael De Vlieger, Oct 30 2017 *)

a(n)=my(k=1);while((2^n-2)*k%n != 0,k++);return(k) \\ Edward Jiang, Sep 09 2014

a(n)=denominator(lift(Mod(2,n)^n-2)/n) \\ Charles R Greathouse IV, Sep 11 2014

a(n) = denominator((2^n - 2)/n). - Juri-Stepan Gerasimov, Sep 09 2014

Extended by Ray Chandler, Apr 11 2009

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

Original entry on oeis.org

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

Alex Ratushnyak, Jul 22 2012

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.

			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.

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

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

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());
    }
  }
}

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

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

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

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

cp's OEIS Frontend

A159353 a(n) = the smallest positive integer such that a(n)*(2^n - 2) is a multiple of n.

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

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