A135406 Sum of squares of gaps between consecutive semiprimes.

4, 13, 14, 30, 31, 67, 68, 77, 78, 127, 128, 129, 138, 139, 188, 197, 201, 217, 221, 222, 238, 247, 263, 288, 297, 322, 331, 332, 333, 349, 353, 354, 355, 476, 501, 517, 526, 527, 531, 532, 533, 569, 585, 586, 635, 636, 637, 641, 642, 723, 732, 733, 737, 762

Offset

1,1

Comments

This is to semiprimes A001358 as A074741 is to primes A000040. What is the semiprime analog of D. R. Heath-Brown's conjecture: Sum_{prime(n)<=N} (prime(n)-prime(n-1))^2 ~ 2*N*log(N) and Marek Wolf's conjecture: Sum_{prime(n)<N} (prime(n)-prime(n-1))^2 = 2*N^2/pi(N) + error term(N), pi(N)=A000720(n).

Links

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Formula

a(n) = SUM[k=1..n] A065516(k)^2 = SUM[k=1..n] (A001358(n+1) - A001358(n))^2.

Example

a(10) = (6-4)^2 + (9-6)^2 + (10-9)^2 + (14-10)^2 + (15-14)^2 + (21-15)^2 + (22-21)^2 + (25-22)^2 + (26-25)^2 + (33-26)^2 = (2^2) + (3^2) + (1^2) + (4^2) + (1^2) + (6^2) + (1^2) + (3^2) + (1^2) + (7^2) = 127.

Maple

A001358 := proc(n) option remember ; local a ; if n = 1 then 4; else for a from A001358(n-1)+1 do if numtheory[bigomega](a) = 2 then RETURN(a) ; fi ; od: fi ; end: A065516 := proc(n) A001358(n+1)-A001358(n) ; end: A135406 := proc(n) add( (A065516(k))^2, k=1..n) ; end: seq(A135406(n), n=1..80) ; # R. J. Mathar, Jan 07 2008

Mathematica

Accumulate[Differences[Select[Range[200], PrimeOmega[#]==2&]]^2] (* Harvey P. Dale, Mar 05 2015 *)

Crossrefs

Cf. A000040, A001358, A065516, A074741.

Sequence in context: A043049 A135465 A135783 * A066825 A014563 A066774

Adjacent sequences: A135403 A135404 A135405 * A135407 A135408 A135409

Keyword

easy,nonn

Author

Jonathan Vos Post, Dec 09 2007

Extensions

More terms from R. J. Mathar, Jan 07 2008

Status

approved