cp's OEIS Frontend

A118780 Semiprime(n)*semiprime(n+3) - semiprime(n+1)*semiprime(n+2), where semiprime(n) is the n-th semiprime.

%I A118780 #19 Sep 08 2024 17:50:44
%S A118780 -14,-6,-5,0,-7,-87,-4,76,-8,-212,64,-4,128,68,-265,31,-12,-177,104,
%T A118780 109,-28,103,-101,-40,-24,-348,-176,253,81,-285,-97,928,364,-841,-257,
%U A118780 -361,-127,-3,-125,603,359,-675,367,-8,-860,139,-3,995,280,-1276,-167,629,145,443,-365,-579,171,-569
%N A118780 Semiprime(n)*semiprime(n+3) - semiprime(n+1)*semiprime(n+2), where semiprime(n) is the n-th semiprime.
%C A118780 Semiprime analog of A117301.
%C A118780 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
%H A118780 Harvey P. Dale, <a href="/A118780/b118780.txt">Table of n, a(n) for n = 1..1000</a>
%F A118780 a(n) = A001358(n)*A001358(n+3) - A001358(n+1)*A001358(n+2).
%e A118780 a(1) = -14 because the determinant of the first block of 4 consecutive semiprimes is:
%e A118780 |4. 6.|
%e A118780 |9. 10|.
%e A118780 a(4) = 0 because the determinant of the 4th block of 4 semiprimes is the first of a presumably infinite number of singular matrices:
%e A118780 |10. 14.|
%e A118780 |15. 21.|.
%e A118780 a(8) = 76, the first positive value in the sequence:
%e A118780 |22. 25.|
%e A118780 |26. 33.|.
%p A118780 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
%t A118780 nmax = 58; spmax = nmax; SP = {};
%t A118780 While[nmax+3 > Length[SP], spmax += nmax; SP = Select[Range[spmax], PrimeOmega[#] == 2&]];
%t A118780 a[n_] := SP[[n]] SP[[n+3]] - SP[[n+1]] SP[[n+2]];
%t A118780 Table[a[n], {n, 1, nmax}] (* _Jean-François Alcover_, Aug 01 2023 *)
%t A118780 #[[1]]#[[4]]-#[[2]]#[[3]]&/@Partition[Select[Range[300],PrimeOmega[#]==2&],4,1] (* _Harvey P. Dale_, Sep 08 2024 *)
%Y A118780 Cf. A001358, A067276, A117301, A118713.
%K A118780 easy,sign
%O A118780 1,1
%A A118780 _Jonathan Vos Post_, May 22 2006
%E A118780 Better definition from _Jens Kruse Andersen_, May 03 2008