A006049 Numbers k such that k and k+1 have the same number of distinct prime divisors.

2, 3, 4, 7, 8, 14, 16, 20, 21, 31, 33, 34, 35, 38, 39, 44, 45, 50, 51, 54, 55, 56, 57, 62, 68, 74, 75, 76, 85, 86, 87, 91, 92, 93, 94, 95, 98, 99, 111, 115, 116, 117, 118, 122, 123, 127, 133, 134, 135, 141, 142, 143, 144, 145, 146, 147, 152, 158, 159, 160, 161, 171, 175

Offset

1,1

Comments

Sequence is infinite, as proved by Schlage-Puchta, who comments: "Buttkewitz found a non-computational proof, and the Goldston-Pintz-Yildirim-sieve yields more precise information". - Charles R Greathouse IV, Jan 09 2013

References

C. Clawson, Mathematical mysteries, Plenum Press 1996, p. 250.

Links

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..2500 from T. D. Noe)

Jan-Christoph Schlage-Puchta, The equation ω(n)=ω(n+1), arXiv:1105.1621 [math.NT], 2011.

Formula

A001221(a(n)) = A001221(a(n)+1). - Reinhard Zumkeller, Jan 22 2013

Mathematica

f[n_] := Length@FactorInteger[n]; t = f /@ Range[175]; Flatten@Position[Rest[t] - Most[t], 0] (* Ray Chandler, Mar 27 2007 *)
Select[Range[200], PrimeNu[#]==PrimeNu[#+1]&] (* Harvey P. Dale, May 09 2012 *)
Flatten[Position[Partition[PrimeNu[Range[200]], 2, 1], _?(#[[1]]==#[[2]]&), {1}, Heads->False]] (* Harvey P. Dale, May 22 2015 *)
SequencePosition[PrimeNu[Range[200]], {x_, x_}][[All, 1]] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Jun 02 2019 *)

Prog

(PARI) is(n)=omega(n)==omega(n+1) \\ Charles R Greathouse IV, Jan 09 2013
(Haskell)
import Data.List (elemIndices)
a006049 n = a006049_list !! (n-1)
a006049_list = map (+ 1) $ elemIndices 0 $
   zipWith (-) (tail a001221_list) a001221_list
-- Reinhard Zumkeller, Jan 22 2013

Crossrefs

Cf. A001221, A052215, A107800, A006073.

Subsequence of A062974.

Sequence in context: A057887 A202116 A215914 * A084541 A240690 A339593

Adjacent sequences: A006046 A006047 A006048 * A006050 A006051 A006052

Keyword

nonn,easy,nice

Author

N. J. A. Sloane

Extensions

Extended by Ray Chandler, Mar 27 2007

Status

approved