A166237 Differences between consecutive products of two distinct primes: a(n) = A006881(n+1) - A006881(n).

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

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

Michael De Vlieger, Table of n, a(n) for n = 1..10000

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. 741-774.

Yang Liu, Peter S. Park, and Zhuo Qun Song, Bounded gaps between products of distinct primes, arXiv:1607.03887 [math.NT], 2016-2017; 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], 2016-2018; Journal of the Mathematical Society of Japan 72:1 (2020), pp. 81-118.

Mathematica

f[n_]:=Last/@FactorInteger[n]=={1, 1}; a=6; lst={}; Do[If[f[n], AppendTo[lst, n-a]; 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(m-1, 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..#T-1] ]; // 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

Added formula to clarify the definition. - N. J. A. Sloane, Jul 19 2022

Status

approved