A140079 Numbers n such that n and n+1 have 5 distinct prime factors.

254540, 310155, 378014, 421134, 432795, 483405, 486590, 486794, 488565, 489345, 507129, 522444, 545258, 549185, 558789, 558830, 567644, 577940, 584154, 591260, 598689, 627095, 634809, 637329, 663585, 666995, 667029, 678755, 687939, 690234

Offset: 1

Author

Artur Jasinski, May 07 2008

Keywords

• nonn

Comments

For the smallest number r such that r and r+1 have n distinct prime factors, see A093548.

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

Subsequence of the variant A321505 defined with "at least 5" instead of "exactly 5" distinct prime factors. See A321495 for the differences. - M. F. Hasler, Nov 12 2018

The subset of numbers where n and n+1 are also squarefree gives A318964. - R. J. Mathar, Jul 15 2023

Links

• Seiichi Manyama, Table of n, a(n) for n = 1..10000
• D. A. Goldston, S. W. Graham, J. Pintz, 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

Cf. A074851, A140077, A140078, A273897.

Cf. A093548.

Equals A321505 \ A321495.

Programs

Mathematica

a = {}; Do[If[Length[FactorInteger[n]] == 5 && Length[FactorInteger[n + 1]] == 5, AppendTo[a, n]], {n, 1, 100000}]; a (*Artur Jasinski*)
Transpose[SequencePosition[Table[If[PrimeNu[n]==5,1,0],{n,700000}],{1,1}]][[1]] (* The program uses the SequencePosition function from Mathematica version 10 *) (* Harvey P. Dale, Jul 25 2015 *)

PARI

is(n)=omega(n)==5 && omega(n+1)==5 \\ Charles R Greathouse IV, Jun 02 2016

Formula

{k: k in A051270 and k+1 in A051270}. - R. J. Mathar, Jul 19 2023