

A166237


Differences between consecutive products of two distinct primes.


11



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, 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, 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
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OFFSET

1,1


COMMENTS

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


LINKS

Table of n, a(n) for n=1..105.
D. A. Goldston, S. W. Graham, J. Pintz and C. Y. Yıldırım, Small gaps between primes and almost primes, arXiv:math/0506067 [math.NT], 2005; Proceedings of the London Mathematical Society 98:3 (May 2009), pp. 741774.
Yang Liu, Peter S. Park, and Zhuo Qun Song, Bounded gaps between products of distinct primes, arXiv:1607.03887 [math.NT], 20162017; Research in Number Theory 3:26 (2017).
Keiju Sono, Small gaps between the set of products of at most two primes, arXiv:1605.02920 [math.NT], 20162018; Journal of the Mathematical Society of Japan 72:1 (2020), pp. 81118.


FORMULA

a(n) = A006881(n+1)  A006881(n).


MATHEMATICA

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


PROG

(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(m1, j, v[j+1]v[j])} \\ Klaus Brockhaus, Oct 13 2009
(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..#T1] ]; // Klaus Brockhaus, Oct 13 2009


CROSSREFS

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).
Sequence in context: A156380 A329708 A263493 * A021878 A247252 A016495
Adjacent sequences: A166234 A166235 A166236 * A166238 A166239 A166240


KEYWORD

nonn


AUTHOR

Vladimir Joseph Stephan Orlovsky, Oct 09 2009


EXTENSIONS

Edited by Klaus Brockhaus, Oct 13 2009


STATUS

approved



