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A166237 Differences between consecutive products of two distinct primes. 11

%I

%S 4,4,1,6,1,4,7,1,1,3,1,7,5,4,2,1,4,3,4,5,3,5,3,1,1,4,2,1,1,11,5,4,3,1,

%T 3,1,6,4,1,7,1,1,2,1,9,3,1,2,5,11,1,5,2,2,7,7,1,1,2,1,3,4,1,1,2,1,1,2,

%U 5,9,2,10,2,4,1,5,3,3,2,7,4,9,4,4,3,1,2,1,1,2,4,5,5,2,2,3,1,2,5,1,4,2,5,9,3

%N Differences between consecutive products of two distinct primes.

%C Goldston, Graham, Pintz & Yıldırım (2005) prove that a(n+1) - a(n) <= 26 infinitely often. They improve this constant to 6 in their 2009 paper. - _Charles R Greathouse IV_, Dec 26 2020

%H D. A. Goldston, S. W. Graham, J. Pintz and C. Y. Yıldırım, <a href="https://arxiv.org/abs/math/0506067">Small gaps between primes and almost primes</a>, arXiv:math/0506067 [math.NT], 2005; Proceedings of the London Mathematical Society 98:3 (May 2009), pp. 741-774.

%H Yang Liu, Peter S. Park, and Zhuo Qun Song, <a href="https://arxiv.org/abs/1607.03887">Bounded gaps between products of distinct primes</a>, arXiv:1607.03887 [math.NT], 2016-2017; Research in Number Theory 3:26 (2017).

%H Keiju Sono, <a href="https://arxiv.org/abs/1605.02920">Small gaps between the set of products of at most two primes</a>, arXiv:1605.02920 [math.NT], 2016-2018; Journal of the Mathematical Society of Japan 72:1 (2020), pp. 81-118.

%F a(n) = A006881(n+1) - A006881(n).

%t f[n_]:=Last/@FactorInteger[n]=={1,1}; a=6;lst={};Do[If[f[n],AppendTo[lst,n-a];a=n],{n,9,6!}];lst

%o (PARI) {m=106; v=vector(m); n=0; c=0; while(c<m, n++; if(bigomega(n)==2&&omega(n)==2, c++; v[c]=n)); w=vector(m-1, j, v[j+1]-v[j])} \\ _Klaus Brockhaus_, Oct 13 2009

%o (MAGMA) T:=[ n: n in [1..360] | #PrimeDivisors(n) eq 2 and &*[ d[2]: d in Factorization(n) ] eq 1 ]; [ T[j+1]-T[j]: j in [1..#T-1] ]; // _Klaus Brockhaus_, Oct 13 2009

%Y Cf. A006881 (products of two distinct primes), A001358 (semiprimes: products of two primes), A065516 (differences between products of two primes), A001223 (differences between consecutive primes).

%K nonn

%O 1,1

%A _Vladimir Joseph Stephan Orlovsky_, Oct 09 2009

%E Edited by _Klaus Brockhaus_, Oct 13 2009

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