A222120 Number of digits in the smallest prime of the form (k+1)^p - k^p, where p = prime(n).

1, 1, 2, 3, 9, 4, 6, 6, 18, 14, 10, 60, 35, 31, 53, 26, 29, 19, 57, 90, 122, 72, 65, 27, 138, 49, 168, 33, 122, 103, 39, 119, 345, 126, 143, 250, 225, 182, 315, 204, 308, 371, 134, 227, 335, 489, 255, 156, 364, 312, 476, 613, 329, 460, 372, 522, 514, 590, 133

Offset: 1

Author

Vladimir Pletser, Feb 07 2013

Keywords

• nonn
• base

Comments

The smallest primes of the form (k+1)^p - k^p are in A121620. The values of k are in A222119. For the primes p = 2, 3, 5, 7, 13, 17, 19, 31, 61, 89, 107, 127, ... (A000043), k = 1 and Mersenne primes 2^p - 1 (A000668) are obtained.

Examples

			a(5) = 9 because the 5th prime is 11, and the smallest prime of the form (k+1)^11 - k^11 is 6^11 - 5^11 = 313968931, which has 9 digits
		

Crossrefs

Cf. A222119 (values of k).

Programs

Mathematica

Table[p = Prime[n]; k = 1; While[q = (k + 1)^p - k^p; ! PrimeQ[q], k++]; Length[IntegerDigits[q]], {n, 60}] (* T. D. Noe, Feb 12 2013 *)

PARI

a222120(n) = {local(p,k); p=prime(n); while(!isprime((k+1)^p - k^p), k=k+1); ceil(log((k+1)^p - k^p)/log(10))} \\ Michael B. Porter, Feb 12 2013