OFFSET
1,1
COMMENTS
By Euclid's argument, the a(i) are distinct.
One can ask whether all primes occur in this sequence.
LINKS
Max Alekseyev, Table of n, a(n) for n = 1..2500
Andrew R. Booker, A variant of the Euclid-Mullin sequence containing every prime, arXiv preprint arXiv:1605.08929 [math.NT], 2016.
FORMULA
For any n, we have Legendre symbol (-a(1)*a(2)*...*a(n-1) / a(n)) = 1. If p is the smallest prime such that (-a(1)*a(2)*...*a(n-1) / p) = 1, then a(n) >= p. Conjecture: For all n, a(n) = p. Note that if b is such that b^2 == -a(1)*a(2)*...*a(n-1) (mod p) and for some I, b == prod_{i in I} a(i) (mod p), then a(n) = p. Heuristically, I must exist for large enough n, since the number of possible subsets I is much larger than p. - Max Alekseyev, Nov 11 2009, May 20 2015
EXAMPLE
a(4)=11 which is the smallest prime dividing the 4 partitions 2+3*5=17, 3+2*5=13, 5+2*3=11, 1+2*3*5=31.
MAPLE
with(numtheory):p:=proc(N) local S, d : S:=NULL:for d in divisors(N) while d^2<=N do S:=S, divisors(d+N/d)[2] od : return(min(S)) end:
a :=n->if n = 1 then 2 else p(mul(a(i), i = 1 .. n-1)) fi :
seq(a(n), n=1..15);
# Robert FERREOL, Oct 01 2019
MATHEMATICA
p[N_Integer] := Module[{S = {}, d, divisorsList},
For[d = 1, d^2 <= N, d++, If[Divisible[N, d], divisorsList = Divisors[d + N/d];
If[Length[divisorsList] >= 2, AppendTo[S, divisorsList[[2]]]]; ]]; Min[S]];
a[n_Integer] := If[n == 1, 2, p[Times @@ Table[a[i], {i, 1, n - 1}]]];
Table[a[n], {n, 1, 14}] (* Hilko Koning, Oct 30 2024 *)
PROG
(PARI) { A167604_list() = my(a, A, p, b, q, z, m); a = []; A=1; while(1, p=2; while( kronecker(-A, p)!=1, p=nextprime(p+1) ); b=lift(sqrt(-A+O(p))); z=znprimroot(p); m=nextprime(random(10^6)); q=lift(prod(i=1, #a, Mod(1+x^znlog(Mod(a[i], p), z, p-1), (1-x^(p-1))*Mod(1, m)) )); if( polcoeff(q, znlog(Mod(b, p), z, p-1), x)==0, error("conjecture failed mod", m)); a=concat(a, [p]); A*=p; print1(p, ", ") ) } /* Max Alekseyev, May 20 2015 */
CROSSREFS
KEYWORD
nonn,changed
AUTHOR
Kok Seng Chua (chuakokseng(AT)hotmail.com), Nov 07 2009
EXTENSIONS
Edited and extended by Max Alekseyev, Nov 11 2009
STATUS
approved