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A196175 Positions of local minima in A001223.

%I A196175 #28 Oct 30 2018 10:31:02
%S A196175 5,7,10,13,17,20,22,26,28,31,33,35,38,41,43,45,49,52,57,60,64,67,69,
%T A196175 75,78,81,83,85,89,93,95,98,100,104,109,113,116,120,122,126,131,134,
%U A196175 136,138,140,142,144,148,152,155,159,163,167,169
%N A196175 Positions of local minima in A001223.
%C A196175 Or, numbers n such that A001223(n-1)>A001223(n)<A001223(n+1).
%C A196175 We start with A001223:
%C A196175 S1= 1, 2, 2, 4, 2, 4, 2, 4, 6, 2, 6, 4, 2, 4, 6, 6, 2, 6, 4, 2, 6, 4, 6, 8, 4, 2, 4.
%C A196175 Local minima are shown in brackets:
%C A196175 S2= 1, 2, 2, {4,2,4}, {4,2,4}, {6,2,6}, {4,2,4}, 6, {6,2,6}, {4,2,6}, {6,4,6}, 8, {4,2,4}, {4,2,4}, {14,4,6}, {6,2,10};
%C A196175 values of local minima are 2, 2, 2, 2, 2, 2, 4, 2, 2, 4, 2, and positions of local minima in A001223 give this sequence. Note that in the first and second brackets we take A001223(6)=4 twice. Also note that all 2's starting with A001223(5) and so on are local minima but there are many other local minima.
%H A196175 Reinhard Zumkeller, <a href="/A196175/b196175.txt">Table of n, a(n) for n = 1..10000</a>
%H A196175 Zak Seidov, <a href="/A196175/a196175.jpg">Crests and troughs in A001223</a>
%e A196175 n=5 A001223(4)=4, A001223(5)=2, A001223(6)=4, and A001223(5) is the local minimum;
%e A196175 n=38: A001223(38)=4 is the local minimum because A001223(37)=6 and A001223(39)=6 both > A001223(38).
%t A196175 nn = 1001; t = Differences[Prime[Range[nn]]]; t2 = {}; Do[If[t[[n - 1]] > t[[n]] && t[[n]] < t[[n + 1]], AppendTo[t2, {n, t[[n]]}]], {n, 2, nn - 2}]; Transpose[t2][[1]] (* _T. D. Noe_, Dec 27 2011 *)
%o A196175 (Haskell)
%o A196175 a196175 n = a196175_list !! (n-1)
%o A196175 a196175_list = map (+ 2) $ elemIndices True $
%o A196175    zipWith (\x y -> x < 0 && y > 0) a036263_list $ tail a036263_list
%o A196175 -- _Reinhard Zumkeller_, Oct 29 2011
%Y A196175 Cf. A001223 (differences between consecutive primes).
%Y A196175 Cf. A036263 (second differences of primes).
%K A196175 nonn
%O A196175 1,1
%A A196175 _Zak Seidov_, Oct 27 2011