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A014566
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Sierpiński numbers of the first kind: n^n + 1.
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44
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2, 2, 5, 28, 257, 3126, 46657, 823544, 16777217, 387420490, 10000000001, 285311670612, 8916100448257, 302875106592254, 11112006825558017, 437893890380859376, 18446744073709551617, 827240261886336764178, 39346408075296537575425, 1978419655660313589123980
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OFFSET
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0,1
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COMMENTS
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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
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REFERENCES
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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.
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LINKS
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FORMULA
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For n>0, resultant of x^n+1 and nx-1. - Ralf Stephan, Nov 20 2004
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MATHEMATICA
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PROG
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(Maxima) A014566[n]:=if n=0 then 2 else n^n+1$
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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