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A121906 Excess of n-th 3-almost prime A014612 over previous prime.

%I A121906 #14 Oct 25 2024 10:59:08
%S A121906 1,1,1,1,4,5,1,1,1,2,3,5,2,5,1,3,2,3,5,3,1,2,1,2,1,1,3,4,11,12,3,1,8,
%T A121906 9,2,3,1,2,3,4,5,1,2,1,5,7,9,2,8,1,11,1,2,3,5,1,3,4,5,4,1,4,3,5,2,4,2,
%U A121906 1,1,2,3,7,9,3,3,1,5,8,1,2
%N A121906 Excess of n-th 3-almost prime A014612 over previous prime.
%C A121906 a(n) = 1 iff n-th 3-almost prime is of form prime + 1 (not yet in OEIS). See also: A109067 3-almost primes of the form semiprime + 1.
%F A121906 a(n) = Min{A014612(n) - p where p is in A000040 and 1<p < A014612(n)}.
%e A121906 a(1) = 8 - 7 = 1; a(2) = 12 - 11 = 1; a(3) = 18 - 17 = 1;
%e A121906 a(4) = 20 - 19 = 1; a(5) = 27 - 23 = 4; a(6) = 28 - 23 = 5;
%e A121906 a(7) = 30 - 29 = 1.
%p A121906 A014612 := proc(n) option remember; local a; if n = 1 then 8; else for a from procname(n-1)+1 do if numtheory[bigomega](a) = 3 then return a; end if; end do: end if; end proc:
%p A121906 A121906 := proc(n) local t; t := A014612(n) ; t-prevprime(t) ; end proc:
%p A121906 seq(A121906(n),n=1..80) ; # _R. J. Mathar_, Dec 22 2010
%t A121906 lim=335;p3=Select[Range[lim], PrimeOmega[#] == 3 &] ;l3=Length[p3];Table[p3[[n]]-NextPrime[p3[[n]],-1],{n,l3}] (* _James C. McMahon_, Oct 24 2024 *)
%Y A121906 Cf. A000040, A014612, A114403.
%K A121906 nonn
%O A121906 1,5
%A A121906 _Jonathan Vos Post_, Sep 01 2006
%E A121906 a(13) corrected by _R. J. Mathar_, Dec 22 2010