A197072 a(n) = prime(2^(n+1)) - 2*prime(2^n).

-1, 1, 5, 15, 25, 49, 97, 181, 433, 819, 1541, 3147, 6271, 12469, 25087, 49455, 99255, 196057, 391815, 781893, 1555935, 3106511, 6191001, 12351963, 24658715, 49173803, 98136735, 195868789, 391110307, 780774507, 1559147549, 3113261723, 6218243597, 12419791799, 24808942497

Offset

0,3

Comments

The prime number theorem implies prime(n) ~ n log n, which explains that lim_{n->oo} A033844(n+1)/A033844(n) = 2. This motivated the present sequence, which has the same property again, e.g., a(46..55)~[1,2,4,8,16,32,64,128,256,512]*10^14. This can be proved by considering an asymptotic expression for prime(n) involving more terms, which yields a(n) = 2^n*(2*log(2)+2/n+O(1/n^2)).

Links

M. F. Hasler, Table of n, a(n) for n = 0..55

Wikipedia, Prime number theorem.

Formula

a(n) ~ 2^(n+1)*(log(2) + 2/n + O(1/n^2)).

a(n) = A033844(n+1) - 2*A033844(n).

Prog

(PARI) a(n)=prime(2<<n)-2*prime(1<<n)  /* works only up to "primelimit" */
(PARI) a(n)=A033844(n+1)-A033844(n)*2  /* if A033844 is a function */
(PARI) vector(#A033844-1, i, A033844[i+1]-A033844[i]*2)  /* assuming A033844 is defined as a vector - note that indices then start at 1 while offset=0 */

Crossrefs

Sequence in context: A168010 A353613 A229312 * A147495 A147426 A100437

Adjacent sequences: A197069 A197070 A197071 * A197073 A197074 A197075

Keyword

sign

Author

M. F. Hasler, Oct 09 2011

Status

approved