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%I A006049 #48 Sep 17 2024 04:04:03
%S A006049 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 A006049 68,74,75,76,85,86,87,91,92,93,94,95,98,99,111,115,116,117,118,122,
%U A006049 123,127,133,134,135,141,142,143,144,145,146,147,152,158,159,160,161,171,175
%N A006049 Numbers k such that k and k+1 have the same number of distinct prime divisors.
%C A006049 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
%C A006049 The asymptotic density of this sequence is 0 (Erdős, 1936). - _Amiram Eldar_, Sep 17 2024
%D A006049 Calvin C. Clawson, Mathematical mysteries, Plenum Press, 1996, p. 250.
%H A006049 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 A006049 Paul Erdős, <a href="https://doi.org/10.1017/S0305004100019277">On a problem of Chowla and some related problems</a>, Mathematical Proceedings of the Cambridge Philosophical Society, Vol. 32, No. 4 (1936), pp. 530-540; <a href="https://www.renyi.hu/~p_erdos/1936-03.pdf">alternative link</a>.
%H A006049 Jan-Christoph Schlage-Puchta, <a href="https://doi.org/10.1112/S0025579300014820">The equation ω(n)=ω(n+1)</a>, Mathematika, Vol. 50, No. 1-2 (2003), pp. 99-101; <a href="https://arxiv.org/abs/1105.1621">arXiv preprint</a>, arXiv:1105.1621 [math.NT], 2011.
%F A006049 A001221(a(n)) = A001221(a(n)+1). - _Reinhard Zumkeller_, Jan 22 2013
%t A006049 f[n_] := Length@FactorInteger[n];t = f /@ Range[175];Flatten@Position[Rest[t] - Most[t], 0] (* _Ray Chandler_, Mar 27 2007 *)
%t A006049 Select[Range[200],PrimeNu[#]==PrimeNu[#+1]&] (* _Harvey P. Dale_, May 09 2012 *)
%t A006049 Flatten[Position[Partition[PrimeNu[Range[200]],2,1],_?(#[[1]]==#[[2]]&),{1},Heads->False]] (* _Harvey P. Dale_, May 22 2015 *)
%t A006049 SequencePosition[PrimeNu[Range[200]],{x_,x_}][[All,1]] (* Requires Mathematica version 10 or later *) (* _Harvey P. Dale_, Jun 02 2019 *)
%o A006049 (PARI) is(n)=omega(n)==omega(n+1) \\ _Charles R Greathouse IV_, Jan 09 2013
%o A006049 (Haskell)
%o A006049 import Data.List (elemIndices)
%o A006049 a006049 n = a006049_list !! (n-1)
%o A006049 a006049_list = map (+ 1) $ elemIndices 0 $
%o A006049    zipWith (-) (tail a001221_list) a001221_list
%o A006049 -- _Reinhard Zumkeller_, Jan 22 2013
%Y A006049 Cf. A001221, A052215, A107800, A006073, A294277, A294278.
%Y A006049 Subsequence of A062974.
%K A006049 nonn,easy,nice
%O A006049 1,1
%A A006049 _N. J. A. Sloane_
%E A006049 Extended by _Ray Chandler_, Mar 27 2007

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