A053341 -id:A053341 - cp's OEIS frontend

A345706 a(n) is the least exponent k > 0 of the n-th prime such that (Product_{j=1..n-1} prime(j)) * prime(n)^k + 1 is a Euclid-Pocklington prime (A053341).

1, 1, 2, 2, 3, 6, 5, 12, 9, 8, 10, 9, 20, 14, 24, 18, 12, 16, 58, 26, 20, 30, 42, 322, 276, 27, 25, 48, 27, 208, 38, 77, 48, 55, 414, 94, 67, 107, 53, 33, 56, 38, 34, 52, 60, 60, 483, 41, 155, 105, 43, 476, 68, 126, 51, 387, 49, 121, 46, 65, 395, 68, 78, 308

Offset: 1

Amiram Eldar, Jun 24 2021

nonn

The corresponding primes are 3, 7, 151, 1471, 279511, 11149928791, 42638305711, 1129919399332465852111, ...

			a(1) = 1 since prime(1) = 2, 2^1 > 1 and 2 + 1 = 3 is a prime.
a(2) = 1 since prime(2) = 3, 3^1 > 2 and 2*3 + 1 = 7 is a prime.
a(3) = 2 since prime(3) = 5, 5^2 > 2*3 and 2*3*5^2 + 1 = 151 is a prime.

Cf. A053341.

a[n_] := Module[{r = Product[Prime[j], {j, 1, n - 1}], p = Prime[n], k}, k = Max[1, Ceiling @ Log[p, r]]; While[!PrimeQ[r*p^k + 1], k++]; k]; Array[a, 64]

A345707 Primes prime(k) at which A345706(k) attains a record value.

Original entry on oeis.org

2, 5, 11, 13, 19, 41, 47, 67, 89, 149, 211, 331, 433, 677, 859, 1301

Offset: 1

Amiram Eldar, Jun 24 2021

The corresponding records values are 1, 2, 3, 6, 12, 20, 24, 58, 322, 414, 483, 527, 1065, 2597, 3181, 6022, ...

			The first 6 terms of A345706 are 1, 1, 2, 2, 3 and 6. The record values, 1, 2, 3 and 6, occur at k = 1, 3, 5 and 6. The corresponding primes are prime(k) = 2, 5, 11 and 13, the first 4 terms of this sequence.

Cf. A053341, A345706.

f[n_] := Module[{r = Product[Prime[j], {j, 1, n - 1}], p = Prime[n], k}, k = Max[1, Ceiling@Log[p, r]]; While[! PrimeQ[r*p^k + 1], k++]; k]; fm = 0; s = {}; Do[f1 = f[k]; If[f1 > fm, fm = f1; AppendTo[s, Prime[ k]]], {k, 1, 50}]; s

cp's OEIS Frontend

A345706 a(n) is the least exponent k > 0 of the n-th prime such that (Product_{j=1..n-1} prime(j)) * prime(n)^k + 1 is a Euclid-Pocklington prime (A053341).

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