|
| |
|
|
A007996
|
|
Primes that divide at least one term of the sequence f given by f(1) = 2, f(n+1) = f(n)^2-f(n)+1 = A000058(n).
|
|
9
|
|
|
|
2, 3, 7, 13, 43, 73, 139, 181, 547, 607, 1033, 1171, 1459, 1861, 1987, 2029, 2287, 2437, 4219, 4519, 6469, 7603, 8221, 9829, 12763, 13147, 13291, 13999, 15373, 17881, 17977, 19597, 20161, 20479, 20641, 20857, 20929, 21661, 23689, 23773, 27031
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
|
OFFSET
|
1,1
|
|
|
COMMENTS
|
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.
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
Because all terms of the sequence f(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
See A180871(n) for the index of the first term of A000058 (this is one less than the index of the f-sequence) divisible by a(n). - M. F. Hasler, Apr 24 2014
|
|
|
LINKS
|
Max Alekseyev, Table of n, a(n) for n = 1..12046 (first 8181 terms are also given at the Andersen link)
J. K. Andersen, Factorization of Sylvester's sequence
R. Mestrovic, Euclid's theorem on the infinitude of primes: a historical survey of its proofs (300 BC--2012) and another new proof, arXiv preprint arXiv:1202.3670, 2012 - From N. J. A. Sloane, Jun 13 2012
R. W. K. Odoni, On the prime divisors of the sequence w_{n+1} = 1+w_1 ... w_n, J. London Math. Soc. 32 (1985), 1-11.
Filip Saidak, A New Proof of Euclid's Theorem, Amer. Math. Monthly, Dec 2006
Eric Weisstein's World of Mathematics, Sylvester's sequence
|
|
|
MAPLE
|
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;
|
|
|
MATHEMATICA
|
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 *)
|
|
|
PROG
|
(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
|
|
|
CROSSREFS
|
The missing primes form A096264.
Cf. A014546, A091335, A091336.
Cf. A180871 (k such that a(n) divides A000058(k)).
Cf. A323605 (smallest prime dividing A000058(n)).
Sequence in context: A078749 A046062 A096263 * A206579 A349327 A166945
Adjacent sequences: A007993 A007994 A007995 * A007997 A007998 A007999
|
|
|
KEYWORD
|
nonn
|
|
|
AUTHOR
|
Bennett Battaile (bennett.battaile(AT)autodesk.com)
|
|
|
EXTENSIONS
|
More terms from Max Alekseyev, Jan 03 2004
Entry revised by N. J. A. Sloane, Jan 28 2007
Definition corrected (following a remark by Don Reble) by M. F. Hasler, Apr 24 2014
|
|
|
STATUS
|
approved
|
| |
|
|
