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

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, 55, 58, 62, 64, 68, 73, 79, 83, 86, 89, 93, 97, 103, 110, 114, 120, 123, 129, 132, 139, 141, 149, 157, 162, 162, 173, 183, 186, 192, 195, 198, 207, 213, 222, 229

Offset: 1

Author

Reinhard Zumkeller, Nov 01 2011

Keywords

• nonn

Comments

a(n) = A038107(n) - A111208(n).

Examples

			a(10) = #{59,61,67,71,73,79,83,89,97} = 9;
a(11) = #{67,71,73,79,83,89,97,101,103,107,109,113} = 12;
a(12) = #{79,83,89,97,101,103,107,109,113,127,131,137,139} = 13.
		

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 1..1000

Crossrefs

Cf. A010051, A000217, A000290.

Programs

Haskell

a194189 n = sum $ map a010051 [n*(n+1) `div` 2 + 1 .. n^2 - 1]

Mathematica

Table[PrimePi[n^2] - PrimePi[n*(n+1)/2], {n, 100}] (* T. D. Noe, Nov 01 2011 *)
PrimePi[#[[2]]]-PrimePi[#[[1]]]&/@Module[{nn=70},Thread[{Accumulate[ Range[ nn]],Range[nn]^2}]] (* Harvey P. Dale, Aug 21 2019 *)