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A135406 Sum of squares of gaps between consecutive semiprimes. 1
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 (list; graph; refs; listen; history; text; internal format)
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

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Last modified November 28 18:22 EST 2021. Contains 349415 sequences. (Running on oeis4.)