A014566 Sierpiński numbers of the first kind: n^n + 1.

2, 2, 5, 28, 257, 3126, 46657, 823544, 16777217, 387420490, 10000000001, 285311670612, 8916100448257, 302875106592254, 11112006825558017, 437893890380859376, 18446744073709551617, 827240261886336764178, 39346408075296537575425, 1978419655660313589123980

Offset: 0

Author

Eric W. Weisstein

Keywords

• nonn
• easy

Comments

Sierpiński primes of the form n^n + 1 are {2,5,257,...} = A121270. The prime p divides a((p-1)/2) for p = {5,7,13,23,29,31,37,47,53,61,71,...} = A003628 Primes congruent to {5, 7} mod 8. p^2 divides a((p-1)/2) for prime p = {29,37,3373,...}. - Alexander Adamchuk, Sep 11 2006

n divides a(n-1) for even n, or 2n divides a(2n-1). a(2n-1)/(2n) = A124899(n) = {1, 7, 521, 102943, 38742049, 23775972551, 21633936185161, 27368368148803711, 45957792327018709121, ...}. 2^n divides a(2^n-1). A014566[2^n - 1] / 2^n = A081216[2^n - 1] = A122000[n] = {1, 7, 102943, 27368368148803711, 533411691585101123706582594658103586126397951, ...}. p+1 divides a(p) for prime p. a(p)/(p+1) = A056852[n] = {7, 521, 102943, 23775972551, 21633936185161, ...}. p^2 divides a((p-1)/2) for prime p = {29, 37, 3373} = A121999(n). - Alexander Adamchuk, Nov 12 2006

References

• Graham Everest, Alf van der Poorten, Igor Shparlinski and Thomas Ward, Recurrence Sequences, Amer. Math. Soc., 2003; see esp. p. 255.
• Maohua Le, Primes in the sequences n^n+1 and n^n-1, Smarandache Notions Journal, Vol. 10, No. 1-2-3, 1999, pp. 156-157.
• Paulo Ribenboim, The Book of Prime Number Records, 2nd ed. New York: Springer-Verlag, p. 74, 1989.

Links

• M. F. Hasler, Table of n, a(n) for n = 0..100
• Florentin Smarandache, Only Problems, Not Solutions!, Xiquan Publ. Hse., 1990, Problem 17.
• Eric Weisstein's World of Mathematics, Sierpiński Number of the First Kind.

Crossrefs

Cf. A000312, A048861, A121270, A003628, A122000, A081216, A056852, A121999, A124899, A134883.

Programs

Mathematica

a(0) = 2; for n>0 Table[n^n+1,{n,1,20}] (* Alexander Adamchuk, Sep 11 2006 *)

Maxima

A014566[n]:=if n=0 then 2 else n^n+1$
makelist(A014566[n],n,0,30); /* Martin Ettl, Oct 29 2012 */

PARI

A014566(n)=n^n+1 /* M. F. Hasler, Jan 21 2009 */

Formula

For n>0, resultant of x^n+1 and nx-1. - Ralf Stephan, Nov 20 2004

E.g.f.: exp(x) + 1/(1+LambertW(-x)). - Vaclav Kotesovec, Dec 20 2014

Sum_{n>=1} 1/a(n) = A134883. - Amiram Eldar, Nov 15 2020

Extensions

More terms from Erich Friedman