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%I #58 Jun 07 2021 01:14:38
%S 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,
%T 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,
%U 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
%N Smallest odd prime p such that (n+1)^p - n^p is prime.
%C 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, ...}.
%C a(96) = 1307, a(98) = 709.
%C 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
%C a(138) through a(150) are 113, >32401, 3, 7, 3, 8839, 5, 7, 13, 3, 5, 271, 13. - _Robert Price_, Feb 17 2012
%C a(137) = 196873 confirmed by Fischer link; a(139) > 260000. - _Ray Chandler_, Feb 26 2017
%H Robert Price and Ray Chandler, <a href="/A125713/b125713.txt">Table of n, a(n) for n = 1..138</a> (first 136 terms from Robert Price)
%H Richard Fischer, <a href="http://www.fermatquotient.com/PrimSerien/PrimPot.txt">Generalized primes of the form (B+1)^N - B^N.</a>
%Y Cf. A058013 (smallest prime p such that (n+1)^p - n^p is prime).
%Y Cf. A065913 (smallest prime of form (n+1)^k - n^k).
%Y Cf. A121091 (smallest nexus prime of the form n^p - (n-1)^p, where p is odd prime).
%Y Cf. A047845, A014076.
%Y Cf. A062585 (numbers n such that k^n - (k-1)^n is prime, where k is 19).
%Y Cf. A000043, A057468, A059801, A059802, A062572-A062666.
%K hard,nonn
%O 1,1
%A _Alexander Adamchuk_, Dec 01 2006, Feb 15 2007
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