A138067 -id:A138067 - cp's OEIS frontend

A138066 Least k > 0 such that (2n-1)^k + 2 is prime, or 0 if no such number exists.

1, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 11, 0, 1, 1, 0, 2, 1, 0, 1, 1, 0, 1, 113, 0, 1, 7, 0, 1, 1, 0, 3, 1, 0, 1, 1, 0, 12, 1, 0, 1, 3, 0, 1, 255, 0, 8, 1, 0, 1, 1, 0, 1, 1, 0, 1, 3, 0, 2, 15, 0, 2, 1, 0, 1, 23, 0, 1, 1, 0, 4, 3, 0, 1, 1, 0, 3, 1, 0, 136, 1, 0, 1

Offset: 1

Alexander Adamchuk, Mar 02 2008

a(3n+1) = 0 for n > 0.

a(84) > 100000. - Ray Chandler, Aug 10 2011

Cf. A084713 (smallest prime of the form (2n-1)^k + 2, or 0 if no such number exists).
Cf. A138067 (least k > 1 such that (2n-1)^k + 2 is prime, or 0 if no such number exists).
Cf. A051783 (k such that 3^k + 2 is prime).
Cf. A087885 (k such that 5^k + 2 is prime).
Cf. A090649, A109076, A113480, A138048, A138049, A138050, A138051, A087886, A113481.

A253178 Least k>=1 such that 2*A007494(n)^k+1 is prime.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 47, 1, 1, 1, 1, 2, 1, 2, 1, 1, 3, 1, 1, 1, 2729, 1, 1, 2, 1, 2, 175, 1, 1, 1, 1, 1, 1, 3, 3, 3, 43, 1, 1, 2, 1, 1, 3, 2, 1, 1, 3, 1, 11, 1, 1, 4, 1, 2, 1, 1, 3, 2, 1, 1, 1, 1, 192275, 2, 1233, 1, 3, 5, 51, 1, 1, 1, 1, 286, 1, 1, 755, 2, 1, 4, 1, 6, 1, 2

Offset: 1

Eric Chen, Mar 20 2015

If n == 1 (mod 3), then for every positive integer k, 2*n^k+1 is divisible by 3 and cannot be prime (unless n=1). Thus we restrict the domain of this sequence to A007494 (n which is not in the form 3j+1).
Conjecture: a(n) is defined for all n.
a(145) > 200000, a(146) .. a(156) = {1, 1, 66, 1, 4, 3, 1, 1, 1, 1, 6}, a(157) > 100000, a(158) .. a(180) = {2, 1, 2, 11, 1, 1, 3, 321, 1, 1, 3, 1, 2, 12183, 5, 1, 1, 957, 2, 3, 16, 3, 1}.
a(n) = 1 if and only if n is in A144769.

Eric Chen, Table of n, a(n) for n = 1..144

Cf. A138066, A138067, A255707, A250200, A119624, A119591, A040076, A046067.

A007494[n_] := 2n - Floor[n/2];
Table[k=1; While[!PrimeQ[2*A007494[n]^k+1], k++]; k, {n, 1, 144}]

a007494(n) = n+(n+1)>>1;
a(n) = for(k=1, 2^24, if(ispseudoprime(2*a007494(n)^k+1),return(k)));

a(n) = A119624(A007494(n)).

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A138066 Least k > 0 such that (2n-1)^k + 2 is prime, or 0 if no such number exists.

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A253178 Least k>=1 such that 2*A007494(n)^k+1 is prime.

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