A118780 - OEIS

%I #19 Sep 08 2024 17:50:44

%S -14,-6,-5,0,-7,-87,-4,76,-8,-212,64,-4,128,68,-265,31,-12,-177,104,

%T 109,-28,103,-101,-40,-24,-348,-176,253,81,-285,-97,928,364,-841,-257,

%U -361,-127,-3,-125,603,359,-675,367,-8,-860,139,-3,995,280,-1276,-167,629,145,443,-365,-579,171,-569

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

%C Semiprime analog of A117301.

%C 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 Harvey P. Dale, <a href="/A118780/b118780.txt">Table of n, a(n) for n = 1..1000</a>

%F a(n) = A001358(n)*A001358(n+3) - A001358(n+1)*A001358(n+2).

%e a(1) = -14 because the determinant of the first block of 4 consecutive semiprimes is:

%e |4. 6.|

%e |9. 10|.

%e 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 |10. 14.|

%e |15. 21.|.

%e a(8) = 76, the first positive value in the sequence:

%e |22. 25.|

%e |26. 33.|.

%p 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 nmax = 58; spmax = nmax; SP = {};

%t While[nmax+3 > Length[SP], spmax += nmax; SP = Select[Range[spmax], PrimeOmega[#] == 2&]];

%t a[n_] := SP[[n]] SP[[n+3]] - SP[[n+1]] SP[[n+2]];

%t Table[a[n], {n, 1, nmax}] (* _Jean-François Alcover_, Aug 01 2023 *)

%t #[[1]]#[[4]]-#[[2]]#[[3]]&/@Partition[Select[Range[300],PrimeOmega[#]==2&],4,1] (* _Harvey P. Dale_, Sep 08 2024 *)

%Y Cf. A001358, A067276, A117301, A118713.

%K easy,sign

%O 1,1

%A _Jonathan Vos Post_, May 22 2006

%E Better definition from _Jens Kruse Andersen_, May 03 2008