A095741 Number of base-2 palindromic primes (A016041) in range [2^2n,2^(2n+1)].

2, 2, 3, 3, 7, 12, 23, 40, 94, 142, 271, 480, 856, 1721, 3099, 5572, 10799, 20782, 39468, 72672, 139867, 274480, 520376, 986318, 1914097, 3726617, 7107443, 13682325, 26430797, 51412565, 99204128, 190457946, 372035117, 727434192, 1407026351, 2724590109, 5315491839

Offset: 1

Author

Antti Karttunen, Jun 12 2004

Keywords

• nonn
• base

Comments

Note that there are no such primes in any range ]2^(2n-1),2^2n], as all even-length binary palindromes are divisible by 3 (cf. A048702).

The ratio a(n)/A036378(2n) converges as follows: 1, 0.4, 0.230769, 0.069767, 0.051095, 0.025862, 0.014268, 0.007006, 0.00461, 0.00193, 0.00101, 0.000487, 0.000235, 0.000127, 0.000061, 0.000029

Links

• A. Karttunen, J. Moyer, C-program for computing the initial terms of this sequence
• Index entries for sequences related to occurrences of various subsets of primes in range ]2^n,2^(n+1)]

Crossrefs

Bisection of the first diagonal of triangle A095759.

Cf. A095731, A117773.

Programs

Mathematica

palindromicQ[n_, b_:10] := TrueQ[IntegerDigits[n, b] == Reverse[IntegerDigits[n, b]]]; Table[Length[Select[Range[2^(2n), 2^(2n + 1)], palindromicQ[#, 2] && PrimeQ[#] &]], {n, 10}] (* Alonso del Arte, Jan 13 2012 *)

PARI

m=vector(65536);u=vector(#m);u[1]=1;for(b=1,#m-1,c=b;e=2^floor(log(b+.5)/log(2));d=0;u[b+1]=e;while(c>0,d=d+e*(c%2);c=floor(c/2);e=e/2);m[b+1]=d);for(x=0,31,h=0;y=2^x;for(w=y,2*y-1,if(x<16,v1=4*y*w+m[w+1];v2=v1+2*y,w1=floor(w/65536);w2=w-65536*w1;v1=262144*y*w1+4*y*w2+65536*u[w1+1]/u[w2+1]*m[w2+1]+m[w1+1];v2=v1+2*y);if(isprime(v1),h++);if(isprime(v2),h++));print(2*x+3" bits: "h)) \\ Martin Raab, Jan 13 2012

Formula

a(n) = A117773(2*n+1). - Chai Wah Wu, Jul 05 2019

Extensions

a(27)-a(32) from Martin Raab, Oct 20 2015

a(33)-a(35) from Chai Wah Wu, Jul 05 2019

a(36)-a(37) from Chai Wah Wu, Jul 11 2019