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A194189 Number of primes between the n-th triangular number and the n-th square.

%I A194189 #22 Aug 21 2019 14:50:42
%S A194189 0,0,1,2,3,3,6,7,8,9,12,13,15,17,18,22,25,27,30,32,35,38,41,43,48,52,
%T A194189 55,58,62,64,68,73,79,83,86,89,93,97,103,110,114,120,123,129,132,139,
%U A194189 141,149,157,162,162,173,183,186,192,195,198,207,213,222,229
%N A194189 Number of primes between the n-th triangular number and the n-th square.
%C A194189 a(n) = A038107(n) - A111208(n).
%H A194189 Reinhard Zumkeller, <a href="/A194189/b194189.txt">Table of n, a(n) for n = 1..1000</a>
%e A194189 a(10) = #{59,61,67,71,73,79,83,89,97} = 9;
%e A194189 a(11) = #{67,71,73,79,83,89,97,101,103,107,109,113} = 12;
%e A194189 a(12) = #{79,83,89,97,101,103,107,109,113,127,131,137,139} = 13.
%t A194189 Table[PrimePi[n^2] - PrimePi[n*(n+1)/2], {n, 100}] (* _T. D. Noe_, Nov 01 2011 *)
%t A194189 PrimePi[#[[2]]]-PrimePi[#[[1]]]&/@Module[{nn=70},Thread[{Accumulate[ Range[ nn]],Range[nn]^2}]] (* _Harvey P. Dale_, Aug 21 2019 *)
%o A194189 (Haskell)
%o A194189 a194189 n = sum $ map a010051 [n*(n+1) `div` 2 + 1 .. n^2 - 1]
%Y A194189 Cf. A010051, A000217, A000290.
%K A194189 nonn
%O A194189 1,4
%A A194189 _Reinhard Zumkeller_, Nov 01 2011