A232190 - OEIS

%I #14 Mar 28 2015 22:38:28

%S 5,9,7,10,11,10,10,13,14,14,15,12,13,11,12,15,18,15,15,15,17,17,18,12,

%T 15,14,14,12,16,14,13,14,16,23,20,16,18,16,17,16,17,16,16,13,17,15,15,

%U 15,20,18,20,19,17,18,18,14,15,18,18,13,17,14,15,17,17,16

%N a(n) is the number of primes of the form 2^b + 2n +- 2^k +- 1 and 2^(b+2) - 2^b - 2n +- 2^k +- 1, where b is the length of the binary representation of 2n, and 0<k<=b.

%C Tested up to n = 1000000000, a(n)> 0.

%C If any zero terms exist, it is likely that the first one will appear in the interval [2*10^9, 2*10^10].

%C The terms of this sequence form a bell-shaped distribution with the commonest value of 21 when n is large enough. Up to the first 100 million terms, the range of a(n) is [3..55].

%H Lei Zhou, <a href="/A232190/b232190.txt">Table of n, a(n) for n = 1..10000</a>

%e When n=1, 2n=2, b=2, the set of numbers of the form 2^b + 2n + 2^k + 1 is {9, 11}; form 2^b + 2n + 2^k - 1: {7, 9}; form 2^b + 2n - 2^k - 1: {1, 3}; form 2^b + 2n - 2^k + 1: {3, 5}; form 2^(b+2) - 2^b - 2n - 2^k - 1: {7, 5}; form 2^(b+2) - 2^b - 2n - 2^k + 1: {9, 7}; form 2^(b+2) - 2^b - 2n + 2^k + 1: {15, 13}; form 2^(b+2) - 2^b - 2n + 2^k - 1: {13, 11}. The union of the above sets is {1, 3, 5, 7, 9, 11, 13, 15}. Among the 8 numbers, 5 are primes. So a(1)=5.

%e When n=11, using the same rule, the candidate number set is {21, 23, 37, 39, 41, 43, 45, 47, 49, 51, 53, 55, 57, 59, 61, 63, 65, 67, 69, 71, 73, 75, 77, 79, 81, 83, 85, 87, 89, 91, 105, 107}. Among these 32 numbers, 15 are prime: {23, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 107}. So a(11)=15.

%t Table[n2 = 2*n; b = Ceiling[Log[2, n2 + 1]]; sdm = 2^b + n2 - 1;

%t sdp = 2^b + n2 + 1; cset = {}; Do[cpmp = sdm + 2^k; cpmm = sdm - 2^k; cppp = sdp + 2^k; cppm = sdp - 2^k; upl = 2^(b + 2); cset = Join[

%t     cset, {cpmp, upl - cpmp, cpmm, upl - cpmm, cppp, upl - cppp, cppm,

%t       upl - cppm}], {k, 1, b}]; cset = Union[cset];

%t size = Length[cset]; ct = 0;

%t Do[If[PrimeQ[cset[[j]]], ct++], {j, 1, size}]; ct, {n, 1, 66}]

%Y Cf. A196697, A196698.

%K nonn

%O 1,1

%A _Lei Zhou_, Nov 20 2013

%E Edited by _Jon E. Schoenfield_, Mar 28 2015