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 A125713 Smallest odd prime p such that (n+1)^p - n^p is prime. 3

%I

%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

%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 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) is 113,>32401,3,7,3,8839,5,7,13,3,5,271,13. - Robert Price, Feb 17 2012

%H Robert Price, <a href="/A125713/b125713.txt">Table of n, a(n) for n = 1..136</a>

%Y 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.

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