A087139 Least k>1 such that p^k - p^(k-1) + 1 is prime for p = prime(n).

2, 2, 3, 2, 11, 2, 5, 30, 15, 3, 6, 10, 81, 3, 17, 961, 15, 7, 2, 5, 6, 2, 3, 3, 12, 3, 57, 5, 16, 5, 166, 15, 13, 2, 3, 2, 30, 2, 25, 3, 47, 3, 3, 2, 521, 9, 3, 15, 17, 42, 17, 51, 39

Offset: 1

Author

T. D. Noe, Aug 18 2003

Keywords

• more
• nonn

Comments

The next term in this sequence, a(54) for the prime p=251, is greater than 73000.

Is there a prime p such that p^k - p^(k-1) + 1 is composite for all k > 1? For the related question of Sierpinski numbers (n such that n*2^k+1 is composite for all k ), the answer is yes.

If n=251^k-251^(k-1)+1 is prime then k mod 10 = 1,5,7 or 9 because n mod 3 = 0 iff k is even and n mod 11 = 0 iff k mod 5 = 3. More exponents can be cleared this way. - Bernardo Boncompagni, Oct 23 2005

Note that k cannot be 8, 14, 20, ... (i.e. k == 2 mod 6) because then p^2 - p + 1 divides p^k - p^(k-1) + 1. - T. D. Noe, Aug 31 2006

References

• See A087126.

Crossrefs

Cf. A040076 (Sierpinski numbers), A087126 (primes of the form p^k - p^(k-1) + 1).

Cf. A122396.

Programs

Mathematica

lst={}; Do[p=Prime[n]; i=2; While[m=p^i-p^(i-1)+1; !PrimeQ[m], i++ ]; AppendTo[lst, i], {n, 53}]; lst