1,1

By Euclid's argument, the terms are distinct.

If I is restricted to the empty set, we get the original Euclid-Mullin sequence (see A000945).

One can ask whether all primes occur in this sequence.

Initialization of the sequence with a(1) = 2 is not necessary: the formula given for successive terms yields 2 anyway (when starting with an empty sequence). Thus the sequence can be considered free of exceptional initialization. - Peter Munn, Oct 16 2025

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.

OEIS Wiki, Empty product

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

To determine a(n+1) = a(2): we have n = 1, so we need to consider all subsets I of {1..n}, noting that {1..n} = {1}; so these subsets are the empty set, {}, and {1}. When I = {} and I' = {1}, Product_{i in I} a(i) is the empty product, which is 1, and Product_{i in I'} a(i) = a(1) = 2. The evaluation specified for I is the least prime dividing the sum of these products (1 + 2 = 3), which is 3. When I = {1}, the 2 products are interchanged, giving the same value 3. Taking the minimum of both values gives a(2) = 3.

a(4)=11 which is the smallest prime dividing any of the expressions derived from 4 partitions of the set of preceding terms: 2 + 3*5 = 17, 3 + 2*5 = 13, 5 + 2*3 = 11, 1 + 2*3*5 = 31.

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

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 *)

(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 */

A167605 lists such n that the first n terms here constitute a permutation of the first n primes.

Cf. A000945, A060772, A344020.

Sequence in context: A067078 A124561 A355503 * A065510 A006721 A111289

Adjacent sequences: A167601 A167602 A167603 * A167605 A167606 A167607

nonn,changed

Kok Seng Chua (chuakokseng(AT)hotmail.com), Nov 07 2009

Edited and extended by Max Alekseyev, Nov 11 2009

Edited by Peter Munn, Oct 05 2025

approved