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

1,1

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.

Formula

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

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 *)

Prog

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

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.

Sequence in context: A206080 A253939 A116248 * A321504 A318896 A328786

Adjacent sequences: A140075 A140076 A140077 * A140079 A140080 A140081

Keyword

nonn

Author

Artur Jasinski, May 07 2008

Extensions

Link provided by Harvey P. Dale, Jun 21 2013

Status

approved