A056852 -id:A056852 - cp's OEIS frontend

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

Eric W. Weisstein

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

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.

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.

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

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

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

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

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

More terms from Erich Friedman

A124899 Sierpinski quotient ((2n-1)^(2n-1) + 1)/(2n) = A014566(2n-1)/(2n).

Original entry on oeis.org

1, 7, 521, 102943, 38742049, 23775972551, 21633936185161, 27368368148803711, 45957792327018709121, 98920982783015679456199, 265572137199362841880960201, 870019499993663001431459704607, 3416070845000481662841943594125601

Offset: 1

Alexander Adamchuk, Nov 12 2006

nonn

2n divides Sierpinski number A014566(2n-1).
2^n divides A014566(2^n-1); A014566(2^n - 1) / 2^n = A081216(2^n - 1) = A122000(n) = {1, 7, 102943, 27368368148803711, 533411691585101123706582594658103586126397951, ...}.
p+1 divides A014566(p) for prime p; A014566(p)/(p+1) = A056852(n) = {7, 521, 102943, 23775972551, 21633936185161, ...}.
Primes in this sequence are {7, 521, 45957792327018709121}.

Seiichi Manyama, Table of n, a(n) for n = 1..194
Eric Weisstein, World of Mathematics. Sierpinski Numbers of the First Kind.

Cf. A014566 (Sierpinski numbers of the first kind: n^n + 1).

Cf. A056826, A056852, A081216, A122000.

List([1..15],n->((2*n-1)^(2*n-1)+1)/(2*n)); # Muniru A Asiru, Apr 08 2018

seq(((2*n-1)^(2*n-1)+1)/(2*n),n=1..20); # Muniru A Asiru, Apr 08 2018

Mathematica
```
Table[((2n-1)^(2n-1)+1)/(2n),{n,1,20}]
```

a(n) = ((2*n-1)^(2*n-1) + 1)/(2*n); \\ Michel Marcus, Apr 08 2018

a(n) = ((2n-1)^(2n-1) + 1)/(2n) = A014566(2n-1)/(2n).

(2n-1)^(a(n)-1) == 1 (mod a(n)). - Thomas Ordowski, Mar 16 2021

cp's OEIS Frontend

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

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