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A006049 Numbers n such that n and n+1 have the same number of distinct prime divisors. 27

%I

%S 2,3,4,7,8,14,16,20,21,31,33,34,35,38,39,44,45,50,51,54,55,56,57,62,

%T 68,74,75,76,85,86,87,91,92,93,94,95,98,99,111,115,116,117,118,122,

%U 123,127,133,134,135,141,142,143,144,145,146,147,152,158,159,160,161,171,175

%N Numbers n such that n and n+1 have the same number of distinct prime divisors.

%C 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

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

%H Amiram Eldar, <a href="/A006049/b006049.txt">Table of n, a(n) for n = 1..10000</a> (terms 1..2500 from T. D. Noe)

%H Jan-Christoph Schlage-Puchta, <a href="http://arxiv.org/abs/1105.1621">The equation ω(n)=ω(n+1)</a>, arXiv:1105.1621 [math.NT], 2011.

%F A001221(a(n)) = A001221(a(n)+1). - _Reinhard Zumkeller_, Jan 22 2013

%t f[n_] := Length@FactorInteger[n];t = f /@ Range[175];Flatten@Position[Rest[t] - Most[t], 0] (* _Ray Chandler_, Mar 27 2007 *)

%t Select[Range[200],PrimeNu[#]==PrimeNu[#+1]&] (* _Harvey P. Dale_, May 09 2012 *)

%t Flatten[Position[Partition[PrimeNu[Range[200]],2,1],_?(#[[1]]==#[[2]]&),{1},Heads->False]] (* _Harvey P. Dale_, May 22 2015 *)

%t SequencePosition[PrimeNu[Range[200]],{x_,x_}][[All,1]] (* Requires Mathematica version 10 or later *) (* _Harvey P. Dale_, Jun 02 2019 *)

%o (PARI) is(n)=omega(n)==omega(n+1) \\ _Charles R Greathouse IV_, Jan 09 2013

%o (Haskell)

%o import Data.List (elemIndices)

%o a006049 n = a006049_list !! (n-1)

%o a006049_list = map (+ 1) $ elemIndices 0 $

%o zipWith (-) (tail a001221_list) a001221_list

%o -- _Reinhard Zumkeller_, Jan 22 2013

%Y Cf. A001221, A052215, A107800, A006073.

%Y Subsequence of A062974.

%K nonn,easy,nice

%O 1,1

%A _N. J. A. Sloane_.

%E Extended by _Ray Chandler_, Mar 27 2007

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Last modified August 6 16:05 EDT 2020. Contains 336255 sequences. (Running on oeis4.)