A390550 Smallest b >= 2 for which P_{n}(b) = Sum_{k=1..n} b^(prime(k)-2) is prime or 0 if such a number does not exist.

0, 2, 2, 2, 0, 14, 3, 12, 5, 8, 18, 360, 71, 40, 9, 4, 210, 68, 13, 970, 174, 292, 169, 14, 264, 10, 705, 34, 64, 390, 8, 3090, 5, 106, 793, 444, 913, 664, 6, 12, 13, 1052, 30, 1432, 699, 1196, 135, 366, 1074, 64, 2, 2602, 198, 162, 572, 4020, 8, 344, 87, 192

Offset

1,2

Comments

Polynomials P_{n}(b) are verified irreducible in Z[b] for n <= 149 with the only exception of P_{5}(b) = 1 + b + b^3 + b^5 + b^9 which admits the nontrivial factorization P_{5}(b) = (b^2 - b + 1) * (b^7 + b^6 - b^4 + b^2 + 2*b + 1) over Z[b].

Conjecture: the polynomials P_{n}(b) are irreducible in Z[b] for all but finitely many values of n.

For all but finitely many n there exists a(n) >= 2 such that P_{n}(a(n)) is prime, verified for n <= 200, with the only exceptions being n = 1 and n = 5.

Links

Davide Oliveri, Table of n, a(n) for n = 1..200

Example

a(5) = 0 in fact P_{5}(b) is reducible over Z[b].
a(6) = 14 in fact P_{6}(b) = 1 + b + b^3 + b^5 + b^9 + b^11. For 2 <= b < 14, P_{6}(b) is composite, while P_{6}(14) = 4070226757031 is prime.

Crossrefs

Cf. A010051, A135482, A390108.

Sequence in context: A265648 A181230 A262372 * A390108 A392180 A292520

Adjacent sequences: A390547 A390548 A390549 * A390551 A390552 A390553

Keyword

nonn

Author

Davide Oliveri, Dec 17 2025

Status

approved