A125713 Smallest odd prime p such that (n+1)^p - n^p is prime.

3, 3, 3, 3, 5, 3, 7, 7, 3, 3, 3, 17, 3, 3, 43, 5, 3, 1607, 5, 19, 127, 229, 3, 3, 3, 13, 3, 3, 149, 3, 5, 3, 23, 3, 5, 83, 3, 3, 37, 7, 3, 3, 37, 5, 3, 5, 58543, 3, 3, 7, 29, 3, 479, 5, 3, 19, 5, 3, 4663, 54517, 17, 3, 3, 5, 7, 3, 3, 17, 11, 47, 61, 19, 23, 3, 5, 19, 7, 5, 7, 3, 3

Offset: 1

Author

Alexander Adamchuk, Dec 01 2006, Feb 15 2007

Keywords

• hard
• nonn

Comments

Corresponding smallest primes of the form (n+1)^p - n^p, where p = a(n) is an odd prime, are listed in A121091(n+1) = {7, 19, 37, 61, 4651, 127, 1273609, 2685817, 271, 331, 397, 6431804812640900941, 547, 631, ...}. a(n) = A058013(n) for n = {4, 7, 10, 12, 13, 16, 17, 19, 22, 24, 25, 27, 28, 31, ...} = A047845(n) = (n-1)/2, where n runs through odd nonprimes (A014076), for n>1. a(97) = 7. a(99)..a(112) = {5, 43, 5, 13, 7, 5, 3, 6529, 59, 3, 5, 5, 113, 5}. a(114) = 139. a(117)..a(129) = {7, 13, 3, 5, 5, 7, 3, 5167, 3, 41, 59, 3, 3}. a(131) = 101. a(n) is currently unknown for n = {113, 115, 116, 130, 132, ...}.

a(96) = 1307, a(98) = 709.

a(137) is probably 196873 from a prime of this form discovered by Jean-Louis Charton in December 2009 and reported to Henri Lifchitz's PRP Top. - Robert Price, Feb 17 2012

a(138) through a(150) are 113, >32401, 3, 7, 3, 8839, 5, 7, 13, 3, 5, 271, 13. - Robert Price, Feb 17 2012

a(137) = 196873 confirmed by Fischer link; a(139) > 260000. - Ray Chandler, Feb 26 2017

Links

• Robert Price and Ray Chandler, Table of n, a(n) for n = 1..138 (first 136 terms from Robert Price)
• Richard Fischer, Generalized primes of the form (B+1)^N - B^N.

Crossrefs

Cf. A058013 (smallest prime p such that (n+1)^p - n^p is prime).

Cf. A065913 (smallest prime of form (n+1)^k - n^k).

Cf. A121091 (smallest nexus prime of the form n^p - (n-1)^p, where p is odd prime).

Cf. A047845, A014076.

Cf. A062585 (numbers n such that k^n - (k-1)^n is prime, where k is 19).

Cf. A000043, A057468, A059801, A059802, A062572-A062666.