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%I #40 Nov 15 2020 03:50:35
%S 2,2,5,28,257,3126,46657,823544,16777217,387420490,10000000001,
%T 285311670612,8916100448257,302875106592254,11112006825558017,
%U 437893890380859376,18446744073709551617,827240261886336764178,39346408075296537575425,1978419655660313589123980
%N Sierpiński numbers of the first kind: n^n + 1.
%C 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
%C 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
%D Graham Everest, Alf van der Poorten, Igor Shparlinski and Thomas Ward, Recurrence Sequences, Amer. Math. Soc., 2003; see esp. p. 255.
%D 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.
%D Paulo Ribenboim, The Book of Prime Number Records, 2nd ed. New York: Springer-Verlag, p. 74, 1989.
%H M. F. Hasler, <a href="/A014566/b014566.txt">Table of n, a(n) for n = 0..100</a>
%H Florentin Smarandache, <a href="http://www.gallup.unm.edu/~smarandache/OPNS.pdf">Only Problems, Not Solutions!</a>, Xiquan Publ. Hse., 1990, Problem 17.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/SierpinskiNumberoftheFirstKind.html">Sierpiński Number of the First Kind</a>.
%F For n>0, resultant of x^n+1 and nx-1. - _Ralf Stephan_, Nov 20 2004
%F E.g.f.: exp(x) + 1/(1+LambertW(-x)). - _Vaclav Kotesovec_, Dec 20 2014
%F Sum_{n>=1} 1/a(n) = A134883. - _Amiram Eldar_, Nov 15 2020
%t a(0) = 2; for n>0 Table[n^n+1,{n,1,20}] (* _Alexander Adamchuk_, Sep 11 2006 *)
%o (PARI) A014566(n)=n^n+1 /* _M. F. Hasler_, Jan 21 2009 */
%o (Maxima) A014566[n]:=if n=0 then 2 else n^n+1$
%o makelist(A014566[n],n,0,30); /* _Martin Ettl_, Oct 29 2012 */
%Y Cf. A000312, A048861, A121270, A003628, A122000, A081216, A056852, A121999, A124899, A134883.
%K nonn,easy
%O 0,1
%A _Eric W. Weisstein_
%E More terms from _Erich Friedman_
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