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A176486 Numbers n such that semiprime(n)/prime(k)=prime and semiprime(n+1)/prime(k+1)=prime.

%I A176486 #9 Feb 04 2019 01:20:52
%S A176486 1,2,3,4,5,6,7,10,14,18,21,29,35,36,39,41,42,45,52,58,59,62,71,73,87,
%T A176486 91,96,97,104,116,120,127,137,141,142,156,168,169,170,178,179,181,185,
%U A176486 188,204,211,227,245,246,249,250,254,255,261,263,279,281,285,290,297,305
%N A176486 Numbers n such that semiprime(n)/prime(k)=prime and semiprime(n+1)/prime(k+1)=prime.
%C A176486 Indices n such that the (n+1)st semiprime has a prime factor which is the next prime after one of the prime factors of the n-th semiprime. - _R. J. Mathar_, Apr 20 2010
%H A176486 Harvey P. Dale, <a href="/A176486/b176486.txt">Table of n, a(n) for n = 1..1000</a>
%e A176486 a(1)=1 because semiprime(1)/prime(1)=2 and semiprime(2)/prime(2)=2;
%e A176486 a(2)=2 because semiprime(2)/prime(1)=3 and semiprime(3)/prime(2)=3;
%e A176486 a(3)=3 because semiprime(3)/prime(2)=3 and semiprime(4)/prime(3)=2.
%p A176486 From _R. J. Mathar_, Apr 20 2010: (Start)
%p A176486 isA001358 := proc(n) numtheory[bigomega](n) = 2 ; end proc:
%p A176486 A001358 := proc(n) option remember ; if n = 1 then return 4 ; else for a from procname(n-1)+1 do if isA001358(a) then return a; end if; end do; end if; end proc:
%p A176486 A084126 := proc(n) min(op(numtheory[factorset](A001358(n)))) ; end proc:
%p A176486 A084127 := proc(n) max(op(numtheory[factorset](A001358(n)))) ; end proc:
%p A176486 A176486 := proc(n) if n = 1 then 1; else for a from procname(n-1)+1 do spl := A084126(a) ; sph := A084127(a) ; sp2l := A084126(a+1) ; sp2h := A084127(a+1) ; if sp2l = nextprime(spl) or sp2h = nextprime(spl) or sp2l = nextprime(sph) or sp2h = nextprime(sph) then return a; end if; end do: end if; end proc:
%p A176486 seq(A176486(n),n=1..80) ; (End)
%t A176486 sppQ[{a_,b_}]:=Module[{t=NextPrime[Transpose[FactorInteger[a]][[1]]],c,d}, c=t[[1]];d=If[Length[t]>1,t[[2]],t[[1]]];Divisible[b,c]|| Divisible[ b,d]]; Flatten[ Position[Partition[Select[Range[1500],PrimeOmega[#] == 2&],2,1],_?sppQ]] (* _Harvey P. Dale_, Mar 16 2015 *)
%Y A176486 Cf. A084126, A084127.
%K A176486 nonn
%O A176486 1,2
%A A176486 _Juri-Stepan Gerasimov_, Apr 18 2010
%E A176486 Corrected (59, 137, 142 inserted, 147 removed) and extended by _R. J. Mathar_, Apr 20 2010