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A344020
A sequence of prime numbers: a(1)=2, a(n+1) is the least prime dividing Product_{i in S} a(i)^2 + Product_{i not in S} a(i)^2, minimized over all subsets S of {1..n}.
2
2, 5, 29, 17, 41, 13, 37, 53, 61, 97, 101, 73, 89, 109, 149, 137, 113, 173, 181, 157, 229, 197, 241, 257, 233, 193, 277, 269, 349, 317, 337, 293, 281, 313, 353, 373, 389, 409, 421, 397, 457, 461, 401, 433, 521, 509, 449, 541, 569, 557, 701, 593, 613, 653, 641, 617, 577, 661, 673, 709, 677, 601, 761, 733, 757, 769, 773, 797
OFFSET
1,1
COMMENTS
A variant of Euclid-Mullin (A000945) and Chua's adaptation (A167604).
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
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
Sequence in context: A358444 A327345 A049050 * A178322 A165161 A098858
KEYWORD
nonn
AUTHOR
Lucas Hoogendijk, May 06 2021
EXTENSIONS
More terms from Max Alekseyev, Jul 03 2022
STATUS
approved