A255707 -id:A255707 - cp's OEIS frontend

A250200 Least number k>1 such that (2n-1)^k - 2 is prime, or 0 if no such number exists.

0, 2, 2, 2, 2, 4, 2, 2, 6, 2, 2, 24, 7, 2, 2, 3, 2, 2, 2, 4, 4, 2, 11, 2, 2, 8, 4, 2, 12, 4, 2, 2, 8, 3, 2, 2, 4, 2, 2, 38, 130, 4, 4, 4, 2, 3, 2, 4, 747, 3, 4, 2, 10, 2, 3, 17, 10, 13, 2, 2, 2, 6, 42, 2, 3, 2, 6, 2, 10, 2, 4, 4, 2, 16, 50, 3, 9, 2, 22, 25

Offset: 1

Robert Price, Mar 02 2015

nonn

Robert Price, Table of n, a(n) for n = 1..143

Cf. A084712, A084713, A084714, A079706, A138066, A255707.

lst = {0}; For[n = 2, n ≤ 143, n++, For[k = 2, k >= 1, k++, If[PrimeQ[(2*n - 1)^k - 2], AppendTo[lst, k]; Break[]]]]; lst
lnk[n_]:=Module[{k=2,c=2n-1},While[!PrimeQ[c^k-2],k++];k]; Join[{0}, Array[ lnk,80,2]] (* Harvey P. Dale, Jul 24 2017 *)

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)).

A292201 a(n) is the smallest value c such that prime(n)^c - 2 is prime, where prime(n) is the n-th prime or -1 if no such c exists.

Original entry on oeis.org

2, 2, 1, 1, 4, 1, 6, 1, 24, 2, 1, 2, 4, 1, 2, 4, 4, 1, 3, 2, 1, 38, 4, 2, 747, 4, 1, 2, 1, 10, 2, 2, 10, 1, 50, 1, 22, 38, 12, 2, 40, 1, 2, 1, 164, 1, 2, 2, 12, 1, 2, 2, 1, 8, 2, 18, 22, 1, 3, 10, 1, 2, 102, 4, 1, 13896, 12, 2, 1122, 1

Offset: 1

Michel Marcus, Sep 11 2017

nonn

a(71) > 38000 (if it exists). - Robert Price, Oct 23 2017

			a(1) = 2 because 2^2 - 2 = 2 is prime;
a(2) = 2 because 3^2 - 2 = 7 is prime;
a(3) = 1 because 5^1 - 2 = 3 is prime;
a(4) = 1 because 7^1 - 2 = 5 is prime.
And these are the least exponents to satisfy the requested property.

Carlos Rivera, Puzzle 887. p(n)^c-2 is prime, The Prime Puzzles and Problems Connection.

Subsequence of A255707.

Table[c = 1; While[! PrimeQ[Prime[n]^c - 2], c++]; c, {n, 24}] (* Michael De Vlieger, Sep 11 2017 *)

a(n) = {my(c = 1, p = prime(n)); while(!isprime(p^c-2), c++); c;}

a(66)-a(70) from Robert Price, Oct 23 2017

A332029 a(n) is the least number k > 0 such that n^k - (n mod 2) - 1 is prime, or 0 if no such number exists.

Original entry on oeis.org

0, 2, 2, 1, 1, 1, 1, 1, 1, 0, 4, 1, 1, 1, 1, 0, 6, 1, 1, 1, 1, 0, 24, 1, 1, 0, 2, 0, 2, 1, 1, 1, 1, 0, 2, 0, 2, 1, 1, 0, 4, 1, 1, 1, 1, 0, 2, 1, 1, 0, 8, 0, 4, 1, 1, 0, 12, 0, 4, 1, 1, 1, 1, 0, 8, 0, 3, 1, 1, 0, 2, 1, 1, 1, 1, 0, 2, 0, 38, 1, 1, 0, 4, 1, 1, 0, 4

Offset: 1

Todor Szimeonov, Feb 05 2020

nonn

Cf. A101264, A255707.

For k >= 1, a(2*k+2) = A101264(k), a(2*k-1) = A255707(k). - Jinyuan Wang, Feb 07 2020

a(n) = 0 for n in A238204. - Michel Marcus, Feb 08 2020 [Proof: a(n) = 1 iff n - 1 is a prime because n^k - 1 is divisible by n - 1, where k > 1 and n is an even number greater than 2. But if n is a term in A238204, n - m is prime only for some m >= 3. Therefore, a(n) = 0 for n in A238204. - Jinyuan Wang, Feb 08 2020]

More terms from Jinyuan Wang, Feb 07 2020

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A250200 Least number k>1 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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A292201 a(n) is the smallest value c such that prime(n)^c - 2 is prime, where prime(n) is the n-th prime or -1 if no such c exists.

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A332029 a(n) is the least number k > 0 such that n^k - (n mod 2) - 1 is prime, or 0 if no such number exists.

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