A006049 - OEIS

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

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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

The asymptotic density of this sequence is 0 (Erdős, 1936). - Amiram Eldar, Sep 17 2024

REFERENCES

Calvin 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)

Paul Erdős, On a problem of Chowla and some related problems, Mathematical Proceedings of the Cambridge Philosophical Society, Vol. 32, No. 4 (1936), pp. 530-540; alternative link.

Jan-Christoph Schlage-Puchta, The equation ω(n)=ω(n+1), Mathematika, Vol. 50, No. 1-2 (2003), pp. 99-101; arXiv preprint, 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)