A140078 Numbers k such that k and k+1 have 4 distinct prime factors.

7314, 8294, 8645, 9009, 10659, 11570, 11780, 11934, 13299, 13629, 13845, 14420, 15105, 15554, 16554, 16835, 17204, 17390, 17654, 17765, 18095, 18290, 18444, 18920, 19005, 19019, 19095, 19227, 20349, 20405, 20769, 21164, 21489, 21735

Offset: 1

Author

Artur Jasinski, May 07 2008

Keywords

• nonn

Comments

Goldston, Graham, Pintz, & Yildirim prove that this sequence is infinite. - Charles R Greathouse IV, Jun 02 2016

The subsequence of terms where k and k+1 are also squarefree is A318896. - R. J. Mathar, Jul 15 2023

References

• David Wells, The Penguin Dictionary of Curious and Interesting Numbers (Rev. ed. 1997), p. 161 (entry for 7314).

Links

• Seiichi Manyama, Table of n, a(n) for n = 1..10000
• D. A. Goldston, S. W. Graham, J. Pintz and C. Y. Yildirim, Small gaps between almost primes, the parity problem and some conjectures of Erdos on consecutive integers, arXiv:0803.2636 [math.NT], 2008.

Crossrefs

Similar sequences with k distinct prime factors: A074851 (k=2), A140077 (k=3), this sequence (k=4), A140079 (k=5).

Cf. A093548.

Equals A321504 \ A321494.

Programs

Mathematica

a = {}; Do[If[Length[FactorInteger[n]] == 4 && Length[FactorInteger[n + 1]] == 4, AppendTo[a, n]], {n, 1, 100000}]; a (* Artur Jasinski, May 07 2008 *)
Transpose[Position[Partition[PrimeNu[Range[20000]],2,1],?(#[[1]] == #[[2]] == 4&),{1},Heads->False]][[1]] (* _Harvey P. Dale, Jun 21 2013 *)
SequencePosition[PrimeNu[Range[22000]],{4,4}][[;;,1]] (* Harvey P. Dale, Jun 20 2024 *)

PARI

isok(n) = (omega(n)==4) && (omega(n+1)==4); \\ Michel Marcus, Sep 04 2015

Formula

{k: k in A033993 and k+1 in A033993}. - R. J. Mathar, Jul 19 2023

Extensions

Link provided by Harvey P. Dale, Jun 21 2013