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 A031439 a(0) = 1, a(n) is the greatest prime factor of a(n-1)^2+1 for n > 0. 7
 1, 2, 5, 13, 17, 29, 421, 401, 53, 281, 3037, 70949, 1713329, 1467748131121, 37142837524296348426149, 101591133424866642486477019709, 1650979973845742266714536305651329, 78343914631785958284737, 4029445531112797145738746391569, 350080544438648120162733678636001, 26208090024628793745288451837610346882122253572537, 4717815978577117335515270825550279551117660519482308365269206484133871485221 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS Does this sequence grow indefinitely, or does it cycle? - Franklin T. Adams-Watters, Oct 02 2006 All a(n) except a(0) = 1 belong to A014442(n) = {2, 5, 5, 17, 13, 37, 5, 13, 41, 101, ...} Largest prime factor of n^2 + 1. All a(n) except a(0) = 1 belong to A002313(n) = {2, 5, 13, 17, 29, 37, 41, 53, 61, 73, 89, 97, 101, ...} Primes congruent to 1 or 2 modulo 4; or, primes of form x^2+y^2; or, -1 is a square mod p. All a(n) except a(0) = 1 and a(1) = 2 are the Pythagorean primes A002144(n) = {5, 13, 17, 29, 37, 41, 53, 61, 73, 89, 97, 101, ...} Primes of form 4n+1. - Alexander Adamchuk, Nov 05 2006 Essentially the same as A072268; A072268(n) = A031439(n-1)^2 + 1. - Charles R Greathouse IV, May 08 2009 LINKS Dennis Langdeau, Table of n, a(n) for n = 0..24 Dario A. Alpern, Factorization: Elliptic Curve Method Jason Papadopoulos, Integer Factorization Source Code. EXAMPLE a(16)=A006530(a(15)^2+1)= A006530(101591133424866642486477019709^2+1)= A006530(10320758390549056348725939119133160378521185060950774444682)= A006530(2*29*23201*4645528280970018601*1650979973845742266714536305651329)= 1650979973845742266714536305651329, factorization of A006530(a(15)^2+1) by Dario A. Alpern's program (see link). MATHEMATICA gpf[n_] := FactorInteger[n][[-1, 1]]; a[0] = 1; a[n_] := a[n] = gpf[a[n - 1]^2 + 1]; Table[an = a[n]; Print[an]; an, {n, 0, 21}] (* Jean-François Alcover, Nov 04 2011 *) NestList[FactorInteger[#^2+1][[-1, 1]]&, 1, 21] (* Harvey P. Dale, Jul 04 2013 *) PROG (PARI) gpf(n)=local(pf); pf=factor(n); pf[matsize(pf)[1], 1] vector(20, i, r=if(i==1, 1, gpf(r^2+1))) CROSSREFS Cf. A056650, A003095. Cf. A002144 - Pythagorean primes: primes of form 4n+1; A002313 - Primes congruent to 1 or 2 modulo 4; A014442 - Largest prime factor of n^2 + 1. Cf. also A072268, A056650, A003095, A125256. Sequence in context: A068486 A099332 A279687 * A074856 A087952 A124255 Adjacent sequences:  A031436 A031437 A031438 * A031440 A031441 A031442 KEYWORD nonn,nice AUTHOR EXTENSIONS One more term from Vladeta Jovovic, Nov 26 2001 a(16) from Reinhard Zumkeller, Aug 07 2004 a(17)-a(21) from Richard FitzHugh (fitzhughrichard(AT)hotmail.com), Aug 12 2004 STATUS approved

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Last modified October 20 15:46 EDT 2019. Contains 328267 sequences. (Running on oeis4.)