A176881 - OEIS

%I #16 Mar 05 2024 07:03:07

%S 1,3,5,2,4,9,11,8,15,2,17,10,21,14,6,16,27,29,8,20,35,4,39,12,41,26,6,

%T 28,45,14,51,34,18,57,10,59,38,40,12,65,44,69,2,24,71,26,77,50,16,81,

%U 56,87,58,32,6,95,64,99,22,36,101,8,68,105,38,24,107,70,4,111,42,76,6,80

%N a(n)=p-q for n-th product of 2 distinct primes p and q (q<p).

%C Where products of two distinct primes are in A006881.

%C If Polignac's conjecture is true, then every even positive integer occurs infinitely many times in this sequence.  - _Clark Kimberling_, Apr 25 2016

%H Clark Kimberling, <a href="/A176881/b176881.txt">Table of n, a(n) for n = 1..1000</a>

%e a(1)=1 because 1=3-2 for A006881(1)=6=3*2; a(2)=3 because 3=5-2 for A006881(2)=10=5*2.

%p A006881 := proc(n) if n = 1 then 6; else for a from procname(n-1)+1 do if numtheory[bigomega](a) = 2 and nops(numtheory[factorset](a)) =2 then return a; end if; end do: end if; end proc:

%p A020639 := proc(n) numtheory[factorset](n) ; min(op(%)) ; end proc:

%p A006530 := proc(n) numtheory[factorset](n) ; max(op(%)) ; end proc:

%p for n from 1 to 130 do c := A006881(n) ; printf("%d,",A006530(c)-A020639(c)) ; end do:

%p # _R. J. Mathar_, May 01 2010

%t mx = 350; t = Sort@Flatten@Table[Prime[n]*Prime[m], {n, Log[2, mx/3]}, {m, n + 1, PrimePi[mx/Prime[n]]}]; (* A006881, _Robert G.Wilson v_, Feb 07 2012 *)

%t u = Table[FactorInteger[t[[k]]][[1]], {k, 1, Length[t]}];

%t u1 = Table[u[[k]][[1]], {k, 1, Length[t]}]  (* A096916 *)

%t PrimePi[u1]  (* A270650 *)

%t v = Table[FactorInteger[t[[k]]][[2]], {k, 1, Length[t]}];

%t v1 = Table[v[[k]][[1]], {k, 1, Length[t]}]  (* A070647 *)

%t PrimePi[v1]  (* A270652 *)

%t d = v1 - u1  (* A176881 *)  (* _Clark Kimberling_, Apr 25 2016 *)

%Y Cf. A006881, A096916, A070647.

%K nonn

%O 1,2

%A _Juri-Stepan Gerasimov_, Apr 27 2010

%E Entries checked by _R. J. Mathar_, May 01 2010