A230769 - OEIS

%I #33 Jan 21 2023 02:20:22

%S 1,2,3,4,5,9,14,15,16,27,45,122,125,213,242,256,263,290,855,1059,2273,

%T 3945,3999,9512,14127,16486,20056,28834,41493,159147,227139,587823

%N Numbers k such that (k+1)*2^k - 1 is prime.

%C 1, 2 and 5 are the only terms of this sequence which are also in A029544. - _Gerasimov Sergey_, Feb 23 2014

%C The next term with this property is > 10000. - _Michael B. Porter_, Feb 23 2014

%C The probability of a given number N being a twin prime grows like 1/(log(N))^2, so for a given n, the probability that it has this property is 1/n^2, and the sum converges.  Are there any n for which n*2^n-1 and n*2^n+1 are both prime? - _Michael B. Porter_, Feb 25 2014

%C We can write (k+1)*2^k - 1 = {(k+1)/2}*4^{(k+1)/2} - 1, and when k is odd, this takes the form of a generalized Woodall prime (base 4). These are listed in A086661. In other words, {2*A086661 - 1} gives all the odd terms of this sequence. - _Jeppe Stig Nielsen_, Oct 16 2019

%C The largest odd term currently known is 3986381 = 2*A086661(21) - 1. - _Jeppe Stig Nielsen_, Oct 16 2019

%H Steven Harvey, <a href="http://harvey563.tripod.com/NearCullen_WoodallPrimes.txt">NearCullen and NearWoodall Primes</a>

%H Prime-Wiki, <a href="https://www.rieselprime.de/ziki/Gen._Woodall_prime_P_2">Near Generalized Woodall primes of the form (n+1)*2^n - 1</a>

%o (PARI) is_A230769(n) = ispseudoprime((n+1)*2^n-1) \\ _M. F. Hasler_, Mar 01 2014

%Y Cf. A029544, A086661, A196273, A236752.

%K nonn,more,hard

%O 1,2

%A _Zak Seidov_, Feb 23 2014

%E Edited and extended to values > 2273 by _M. F. Hasler_, Mar 01 2014

%E More terms from _Jeppe Stig Nielsen_, Oct 16 2019