A225101 Numerator of (2^n - 2)/n.

0, 1, 2, 7, 6, 31, 18, 127, 170, 511, 186, 2047, 630, 8191, 10922, 32767, 7710, 131071, 27594, 524287, 699050, 2097151, 364722, 8388607, 6710886, 33554431, 44739242, 19173961, 18512790, 536870911, 69273666, 2147483647, 2863311530, 8589934591, 34359738366, 34359738367, 3714566310

Offset: 1

Author

Alonso del Arte, Apr 28 2013

Keywords

• easy
• nonn
• frac

Comments

That (2^n - 2)/n is an integer when n is prime can easily be proved as a simple consequence of Fermat's little theorem.

It was believed long ago that (2^n - 2)/n is an integer only when n = 1 or a prime. In 1819, Frédéric Sarrus found the smallest counterexample, 341; these pseudoprimes are now sometimes called "Sarrus numbers" (A001567).

Examples

			a(4) = 7 because (2^4 - 2)/4 = 7/2.
a(5) = 6 because (2^5 - 2)/5 = 6.
a(6) = 31 because (2^6 - 2)/6 = 31/3.
		

References

• Alkiviadis G. Akritas, Elements of Computer Algebra With Application. New York: John Wiley & Sons (1989): 66.
• George P. Loweke, The Lore of Prime Numbers. New York: Vantage Press, 1982, p. 22.

Links

• Colin Barker, Table of n, a(n) for n = 1..1000
• Eric Weisstein's World of Mathematics, Chinese Hypothesis

Crossrefs

Cf. A001567, A064535, A159353 (denominators).

Programs

Magma

[Numerator((2^n - 2)/n): n in  [1..60]]; // Vincenzo Librandi, Nov 09 2014

Maple

A225101:=n->numer((2^n-2)/n): seq(A225101(n), n=1..50); # Wesley Ivan Hurt, Nov 10 2014

Mathematica

Table[Numerator[(2^n - 2)/n], {n, 50}]

PARI

vector(100, n, numerator((2^n - 2)/n)) \\ Colin Barker, Nov 09 2014