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%I #38 Jan 17 2024 05:08:12
%S 0,1,1,2,3,4,5,6,6,6,7,8,9,10,10,10,11,12,13,14,14,14,15,16,16,16,16,
%T 16,17,18,19,20,20,20,20,20,21,22,22,22,23,24,25,26,26,26,27,28,28,28,
%U 28,28,29,30,30,30,30,30,31,32,33,34,34,34,34,34,35,36,36,36,37,38,39
%N Counts transitions between prime and nonprime to reach the number n.
%C The following sequences all appear to have the same parity (with an extra zero term at the start of A010051): A010051, A061007, A035026, A069754, A071574. - _Jeremy Gardiner_, Aug 09 2002
%H T. D. Noe, <a href="/A069754/b069754.txt">Table of n, a(n) for n=1..1000</a>
%F When n is prime, a(n) = 2*pi(n) - 3. When n is composite, a(n) = 2*pi(n) - 2. pi(n) is the prime counting function A000720.
%F For n > 2: a(n) = 2*A000720(n) - 2 - A010051(n). - _Reinhard Zumkeller_, Dec 04 2012
%e a(6) = 4 because there are 4 transitions: 1 to 2, 3 to 4, 4 to 5 and 5 to 6.
%t For[lst={0}; trans=0; n=2, n<100, n++, If[PrimeQ[n]!=PrimeQ[n-1], trans++ ]; AppendTo[lst, trans]]; lst
%t (* Second program: *)
%t pts[n_]:=Module[{c=2PrimePi[n]},If[PrimeQ[n],c-3,c-2]]; Join[{0,1},Array[ pts,80,3]] (* _Harvey P. Dale_, Nov 12 2011 *)
%t Accumulate[If[Sort[PrimeQ[#]]=={False,True},1,0]&/@Partition[ Range[ 0,80],2,1]] (* _Harvey P. Dale_, May 06 2013 *)
%o (Haskell)
%o a069754 1 = 0
%o a069754 2 = 1
%o a069754 n = 2 * a000720 n - 2 - (toInteger $ a010051 $ toInteger n)
%o -- _Reinhard Zumkeller_, Dec 04 2012
%Y Cf. A000720 (pi).
%Y Cf. A211005 (run lengths).
%Y Same parity: A010051, A061007, A035026, A071574.
%K easy,nice,nonn
%O 1,4
%A _T. D. Noe_, May 02 2002
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