OFFSET
1,1
COMMENTS
All a(i) must be unique and, apart from 2, must be congruent to 1 (mod 4) as p only divides Product_{i in S} a(i)^2 + Product_{i not in S} a(i)^2 if -1 is a quadratic residue modulo p.
Whether all primes congruent to 1 (mod 4) occur in this sequence is unknown.
For n > 1, a(n) >= p, where p is the smallest prime p such that p == 1 (mod 4) and a(2)*a(3)*...*a(n-1) is a nonzero square modulo p. Conjecture: a(n) = p. - Jinyuan Wang and Max Alekseyev, Jul 04 2022
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.
Lucas M. H. Hoogendijk, Prime Generators, UU bachelor thesis, 2020.
Lucas Hoogendijk, Python code used to compute the first 27 terms (all known terms at the time of upload).
EXAMPLE
For n=4 we obtain the 4 partitions with their products: 1 + 2^2 * 5^2 * 29^2 = 84101 = 37 * 2273, 2^2 + 5^2 * 29^2 = 21029 = 17*1237, 5^2 + 2^2 * 29^2 = 3389 and 2^2 * 5^2 + 29^2 = 941. The minimum of the primes dividing these is 17, thus a(4)=17.
PROG
(PARI) { A344020_list() = my(a, A, m, p, b, q, z); print1(2, ", "); a = [2]; A=1; while(1, p=5; while( kronecker(A, p)!=1 || p%4!=1, p=nextprime(p+1) ); b=lift(sqrt(A+O(p))*(1+sqrt(-1+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 && 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, Jul 04 2022
CROSSREFS
KEYWORD
nonn
AUTHOR
Lucas Hoogendijk, May 06 2021
EXTENSIONS
More terms from Max Alekseyev, Jul 03 2022
STATUS
approved