A391020 Numbers k such that no numbers of the form 1 + (product of k distinct primes of first k+1 primes) are prime.

18, 23, 24, 32, 50, 78, 86, 88, 115, 118, 131, 133, 137, 143, 145, 146, 149, 152, 153, 157, 159, 162, 165, 190, 207, 209, 230, 234, 237, 242, 244, 249, 253, 259, 284, 297, 301, 312, 314, 316, 322, 328, 339, 347, 351, 361, 376, 385, 389, 390, 392, 395

Offset

1,1

Comments

While A014545 (k such that k-th Euclid number is prime) is a rare sequence (only 27 primes for k<500000), the situation is different for "relaxed Euclid numbers" of the defined form. For 348 of the first 400 k's (the other 52 are this sequence) there exists at least one relaxed k-th Euclid number that is prime.

Links

Hermann Stamm-Wilbrandt, Table of n, a(n) for n = 1..230 (non-terms 1127, 1447 and 1639 removed from Serge Batalov bfile)

Example

18 is a term because no subset of 18 distinct primes from the first 19 primes [2,3,...,67] multiplied plus 1 is prime.

Prog

(PARI)
isok(n) = {forsubset([n+1, n], s, if(ispseudoprime(1+vecprod([prime(i)|i<-s])), return(0))); 1};
for(n=1, 400, if(isok(n), print1(n", ")))
(PARI) isok2(n) = {my(p=vecprod(primes(n+1))); parfor(i=2, n+1, ispseudoprime(1+p/prime(i)), r, if(r, return(0)); ); 1}; \\ parallel with early abort, Hermann Stamm-Wilbrandt, Feb 10 2026

Crossrefs

Disjoint to A014545.

Sequence in context: A267826 A339473 A361627 * A002505 A182438 A050772

Adjacent sequences: A391017 A391018 A391019 * A391021 A391022 A391023

Keyword

nonn

Author

Hermann Stamm-Wilbrandt, Nov 25 2025

Status

approved