A007996 - OEIS

%I #53 Nov 28 2024 11:14:56

%S 2,3,7,13,43,73,139,181,547,607,1033,1171,1459,1861,1987,2029,2287,

%T 2437,4219,4519,6469,7603,8221,9829,12763,13147,13291,13999,15373,

%U 17881,17977,19597,20161,20479,20641,20857,20929,21661,23689,23773,27031

%N Primes that divide at least one term of Sylvester's sequence s = A000058: s(n+1) = s(n)^2 - s(n) + 1, s(0) = 2.

%C Or, let S_1 = [2] and let S_{n+1} = list formed by sorting the union of S_n together with all prime factors of 1 + Product_i S_n(i) into increasing order; sequence is limit as n -> infinity of S_n.

%C Prime divisors of the terms of Sylvester's sequence A000058. - _Max Alekseyev_, Jan 03 2004. Also of A007018. - _N. J. A. Sloane_, Jan 27 2007

%C Because all terms of the sequence s(n) are coprime, a prime can divide at most one term. Odoni shows that primes p > 3 in this sequence must satisfy p = 1 (mod 6). - _T. D. Noe_, Sep 25 2010

%C See A180871(n) for the index of the first term of A000058 (this is one less than the index of the s-sequence) divisible by a(n). - _M. F. Hasler_, Apr 24 2014

%H Max Alekseyev, <a href="/A007996/b007996.txt">Table of n, a(n) for n = 1..12046</a> (first 8181 terms are also given at the Andersen link)

%H J. K. Andersen, <a href="http://primerecords.dk/sylvester-factors.htm">Factorization of Sylvester's sequence</a>

%H R. Mestrovic, <a href="http://arxiv.org/abs/1202.3670">Euclid's theorem on the infinitude of primes: a historical survey of its proofs (300 BC--2012) and another new proof</a>, arXiv preprint arXiv:1202.3670, 2012 - From N. J. A. Sloane, Jun 13 2012

%H R. W. K. Odoni, <a href="http://jlms.oxfordjournals.org/content/s2-32/1/1.extract">On the prime divisors of the sequence w_{n+1} = 1+w_1 ... w_n</a>, J. London Math. Soc. 32 (1985), 1-11.

%H Filip Saidak, <a href="http://www.jstor.org/stable/27642094">A New Proof of Euclid's Theorem</a>, Amer. Math. Monthly, Dec 2006

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/SylvestersSequence.html">Sylvester's sequence</a>

%p n := 1; for p do if isprime(p) then x := 2 mod p; S := {}; while not member(x,S) do if x=0 then a[n] := p; n := n+1; break; fi; S := S union {x}; x := (x^2-x+1) mod p; od; fi; od;

%t t={}; p=1; While[Length[t]<100, p=NextPrime[p]; s=Mod[2,p]; k=0; modSet={}; While[s>0 && !MemberQ[modSet,s], AppendTo[modSet,s]; k++; s=Mod[s^2-s+1,p]]; If[s==0, AppendTo[t,{p,k}]]]; Transpose[t][[1]] (* _T. D. Noe_, Sep 25 2010 *)

%o (PARI) is(n)=my(k=Mod(2,n)); for(i=1, n, k=(k-1)*k+1; if(k==0, return(isprime(n)))); n==2 \\ _Charles R Greathouse IV_, Sep 30 2015

%Y The missing primes form A096264.

%Y Cf. A014546, A091335, A091336.

%Y Cf. A180871 (k such that a(n) divides A000058(k)).

%Y Cf. A323605 (smallest prime dividing A000058(n)).

%K nonn

%O 1,1

%A Bennett Battaile (bennett.battaile(AT)autodesk.com)

%E More terms from _Max Alekseyev_, Jan 03 2004

%E Entry revised by _N. J. A. Sloane_, Jan 28 2007

%E Definition corrected (following a remark by _Don Reble_) by _M. F. Hasler_, Apr 24 2014