A140078 - OEIS

%I #37 Jun 20 2024 18:00:59

%S 7314,8294,8645,9009,10659,11570,11780,11934,13299,13629,13845,14420,

%T 15105,15554,16554,16835,17204,17390,17654,17765,18095,18290,18444,

%U 18920,19005,19019,19095,19227,20349,20405,20769,21164,21489,21735

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

%C Goldston, Graham, Pintz, & Yildirim prove that this sequence is infinite. - _Charles R Greathouse IV_, Jun 02 2016

%C The subsequence of terms where k and k+1 are also squarefree is A318896. - _R. J. Mathar_, Jul 15 2023

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

%H Seiichi Manyama, <a href="/A140078/b140078.txt">Table of n, a(n) for n = 1..10000</a>

%H D. A. Goldston, S. W. Graham, J. Pintz and C. Y. Yildirim, <a href="http://arxiv.org/abs/0803.2636">Small gaps between almost primes, the parity problem and some conjectures of Erdos on consecutive integers</a>, arXiv:0803.2636 [math.NT], 2008.

%F {k: k in A033993 and k+1 in A033993}. - _R. J. Mathar_, Jul 19 2023

%t a = {}; Do[If[Length[FactorInteger[n]] == 4 && Length[FactorInteger[n + 1]] == 4, AppendTo[a, n]], {n, 1, 100000}]; a (* _Artur Jasinski_, May 07 2008 *)

%t Transpose[Position[Partition[PrimeNu[Range[20000]],2,1],_?(#[[1]] == #[[2]] == 4&),{1},Heads->False]][[1]] (* _Harvey P. Dale_, Jun 21 2013 *)

%t SequencePosition[PrimeNu[Range[22000]],{4,4}][[;;,1]] (* _Harvey P. Dale_, Jun 20 2024 *)

%o (PARI) isok(n) = (omega(n)==4) && (omega(n+1)==4); \\ _Michel Marcus_, Sep 04 2015

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

%Y Cf. A093548.

%Y Equals A321504 \ A321494.

%K nonn

%O 1,1

%A _Artur Jasinski_, May 07 2008

%E Link provided by _Harvey P. Dale_, Jun 21 2013