|
Search: id:A135406
|
|
|
| A135406 |
|
Sum of squares of gaps between consecutive semiprimes. |
|
+0
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; listen)
|
|
|
OFFSET
|
1,1
|
|
|
COMMENT
|
This is to semiprimes A001358 as A074741 is to primes A000040. What is the semiprime analogue of D. R. Heath-Brown's conjecture: Sum_{ithprime(n)<=N} (ithprime(n)-ithprime(n-1))^2 ~ 2*N*log(N), and Marek Wolf's conjecture: Sum_{ithprime(n)<N} (ithprime(n)-ithprime(n-1))^2 = 2*N^2/pi(N) + error term(N), pi(N)=A000720(n).
|
|
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 (mathar(AT)strw.leidenuniv.nl), Jan 07 2008
|
|
CROSSREFS
|
Cf. A000040, A001358, A065516, A074741.
Adjacent sequences: A135403 A135404 A135405 this_sequence A135407 A135408 A135409
Sequence in context: A043049 A135465 A135783 this_sequence A066825 A014563 A066774
|
|
KEYWORD
|
easy,nonn
|
|
AUTHOR
|
Jonathan Vos Post (jvospost3(AT)gmail.com), Dec 09 2007
|
|
EXTENSIONS
|
More terms from R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Jan 07 2008
|
|
|
Search completed in 0.002 seconds
|
