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A125958 Least number k > 0 such that (2^k + (2n-1)^k)/(2n+1) is prime.

%I A125958 #19 Aug 28 2024 10:55:53
%S A125958 3,3,3,5,3,3,7,3,5,5,11,3,19,11,3,229,47,5,257,3,19,31,17,11,13,3,3,5,
%T A125958 5,59,23,3,3,7,79,3,3373,3,3,7,13,7,7,3527,593,19,3,3,13,13,11,19,41,
%U A125958 3,7,109,3,227,13,5,5,3,239,5,3251,3,1237,3,7,31,3,7
%N A125958 Least number k > 0 such that (2^k + (2n-1)^k)/(2n+1) is prime.
%C A125958 All terms are odd primes.
%C A125958 a(38),...,a(43) = {3,3,7,13,7,7}.
%C A125958 a(46),...,a(64) = {19,3,3,13,13,11,19,41,3,7,109,11,227,13,5,5,3,239,5}.
%C A125958 a(66) = 3. a(68),...,a(72) = {3,7,31,3,7}.
%C A125958 a(74),...,a(92) = {3,5,19,17,3,83,3,3,19,19,11,11,61,3,7,7,3,41,29}.
%C A125958 a(94) = 5. a(97),a(98) = {19,7}. a(100) = 31.
%C A125958 a(n) is currently unknown for n = {37,44,45,65,67,73,93,95,96,99,...}.
%C A125958 From _Kevin P. Thompson_, May 18 2022: (Start)
%C A125958 All known terms from n=1..100 correspond to proven primes.
%C A125958 a(96) > 10250.
%C A125958 a(99) > 10250. (End)
%C A125958 Presuming every prime is seen at least once, one can specifically seek those with fixed k. Doing this, a(174) = 37, a(368) = 43 for example. - _Bill McEachen_, Aug 26 2024
%H A125958 Kevin P. Thompson, <a href="/A125958/b125958.txt">Table of n, a(n) for n = 1..95</a>
%e A125958 For n=4, the expression (2^k + (2n-1)^k)/(2n+1) takes on values 1, 53/9, 39, 2417/9, and 1871 for k=1..5. Since 1871 is the first prime number to occur, a(4) = 5.
%t A125958 Do[k = 1; While[ !PrimeQ[(2^k + (2n-1)^k)/(2n+1)], k++ ]; Print[k], {n, 100}] (* _Ryan Propper_, Mar 29 2007 *)
%Y A125958 Cf. A000978 ((2^n + 1)/3 is prime), A057469 ((2^n + 3^n)/5 is prime).
%Y A125958 Cf. A082387 ((2^n + 5^n)/7 is prime), A125955 ((2^n + 7^n)/9 is prime).
%Y A125958 Cf. A125956 ((2^n + 9^n)/11 is prime), A125955 ((2^n + 11^n)/13 is prime).
%K A125958 nonn,hard
%O A125958 1,1
%A A125958 _Alexander Adamchuk_, Feb 06 2007
%E A125958 More terms from _Ryan Propper_, Mar 29 2007
%E A125958 a(65)-a(72) from _Kevin P. Thompson_, May 18 2022