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A118780
Semiprime(n)*semiprime(n+3) - semiprime(n+1)*semiprime(n+2), where semiprime(n) is the n-th semiprime.
3
-14, -6, -5, 0, -7, -87, -4, 76, -8, -212, 64, -4, 128, 68, -265, 31, -12, -177, 104, 109, -28, 103, -101, -40, -24, -348, -176, 253, 81, -285, -97, 928, 364, -841, -257, -361, -127, -3, -125, 603, 359, -675, 367, -8, -860, 139, -3, 995, 280, -1276, -167, 629, 145, 443, -365, -579, 171, -569
OFFSET
1,1
COMMENTS
Semiprime analog of A117301.
By construction, every entry is also the difference between two 4-almost primes: a(1) = A014613(4)-A014613(5); a(2) = A014613(9)-A014613(11); a(3) = A014613(16)-A014613(18); a(4) = A014613(27)-A014613(27); etc. - R. J. Mathar, Nov 27 2007
LINKS
FORMULA
a(n) = A001358(n)*A001358(n+3) - A001358(n+1)*A001358(n+2).
EXAMPLE
a(1) = -14 because the determinant of the first block of 4 consecutive semiprimes is:
|4. 6.|
|9. 10|.
a(4) = 0 because the determinant of the 4th block of 4 semiprimes is the first of a presumably infinite number of singular matrices:
|10. 14.|
|15. 21.|.
a(8) = 76, the first positive value in the sequence:
|22. 25.|
|26. 33.|.
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: A118780 := proc(n) A001358(n)*A001358(n+3)-A001358(n+1)*A001358(n+2) ; end: seq(A118780(n), n=1..58) ; # R. J. Mathar, Nov 27 2007
MATHEMATICA
nmax = 58; spmax = nmax; SP = {};
While[nmax+3 > Length[SP], spmax += nmax; SP = Select[Range[spmax], PrimeOmega[#] == 2&]];
a[n_] := SP[[n]] SP[[n+3]] - SP[[n+1]] SP[[n+2]];
Table[a[n], {n, 1, nmax}] (* Jean-François Alcover, Aug 01 2023 *)
#[[1]]#[[4]]-#[[2]]#[[3]]&/@Partition[Select[Range[300], PrimeOmega[#]==2&], 4, 1] (* Harvey P. Dale, Sep 08 2024 *)
CROSSREFS
KEYWORD
easy,sign
AUTHOR
Jonathan Vos Post, May 22 2006
EXTENSIONS
Better definition from Jens Kruse Andersen, May 03 2008
STATUS
approved