A390549 Smallest b>=2 for which Q_{n}(b) = prime(n) + Sum{k=1..n-1} b^(prime(n) - prime(k)) is prime, or -1 if no such b exists.

2, 2, 2, 2, 2, 6, 15, 10, 3, 76, 22, 78, 144, 18, 56, 214, 3, 2, 65, 468, 3, 2, 87, 1510, 211, 38, 152, 1332, 213, 22, 765, 3690, 350, 160, 111, 1116, 311, 378, 1135, 1218, 296, 36, 312, 1270, 153, 2, 120, 120, 2002, 16, 869, 4878, 22, 272, 418, 242, 328

Offset

1,1

Comments

Conjecture: Q_{n}(b) takes infinitely many prime values for b >= 2 and n >= 3.

Links

Table of n, a(n) for n=1..57.

Example

For n = 7, prime(7) = 17 and Q_{7}(b) = 17 + b^15 + b^14 + b^12 + b^10 + b^6 + b^4.
For 2 <= b <= 14 all values of Q_{7}(b) are composite while Q_{7}(15) = 467217139405972517 is prime, so a(7) = 15.

Mathematica

qb[n_, b_]:=Prime[n]+Sum[b^(Prime[n]-Prime[k]), {k, 1, n-1}]; a[n_]:=Module[{b=2}, While[!PrimeQ[qb[n, b]], b++]; b]; Array[a, 57] (* James C. McMahon, Nov 16 2025 *)

Crossrefs

Cf. A390108.

Sequence in context: A214080 A278241 A010671 * A339164 A323443 A334511

Adjacent sequences: A390546 A390547 A390548 * A390550 A390551 A390552

Keyword

nonn

Author

Davide Oliveri, Nov 10 2025

Status

approved