A090970 Number of primes strictly between T(n) and T(n+1), where T(n) = n-th triangular number.

1, 1, 1, 2, 2, 1, 2, 3, 2, 2, 3, 3, 3, 3, 2, 4, 3, 3, 4, 4, 4, 4, 4, 4, 4, 4, 5, 5, 6, 4, 5, 3, 6, 6, 7, 5, 5, 6, 4, 8, 5, 6, 6, 8, 6, 8, 5, 7, 5, 11, 4, 6, 9, 7, 8, 9, 8, 7, 7, 9, 7, 8, 7, 12, 5, 9, 9, 11, 9, 7, 7, 12, 10, 10, 9, 9, 9, 6, 11, 10, 11, 9, 12, 11, 12, 9, 10, 11, 12, 10, 13, 9, 11, 10, 12

Offset: 1

Author

Amarnath Murthy, Jan 03 2004

Keywords

• nonn

Examples

			a(8)=3 because between T(8)=36 and T(9)=45 we have the prime numbers 37,41 and 43.
		

Links

• T. D. Noe, Table of n, a(n) for n = 1..10000

Crossrefs

Essentially the same as A065382 and A066888.

Programs

Maple

a:= proc(n) local ct,j: ct:=0: for j from n*(n+1)/2+1 to (n+1)*(n+2)/2-1 do if isprime(j)=true then ct:=ct+1 else ct:=ct fi: od: end: seq(a(n),n=1..103); # Emeric Deutsch, Feb 23 2005

Mathematica

With[{trs=Partition[Accumulate[Range[100]],2,1]},Join[{1},Rest[ PrimePi[ #[[2]]]- PrimePi[#[[1]]]&/@trs]]] (* Harvey P. Dale, Aug 25 2015 *)

Python

from sympy import primerange
def A090970(n): return sum(1 for p in primerange((n*(n+1)>>1)+1,(n+2)*(n+1)>>1)) # Chai Wah Wu, May 22 2025

Extensions

More terms from Emeric Deutsch, Feb 23 2005