A065382 Number of primes between n(n+1)/2 (exclusive) and (n+1)(n+2)/2 (inclusive).

2, 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

Offset: 1

Author

Reinhard Zumkeller, Nov 05 2001

Keywords

• nonn

Comments

Inspired by the weaker Legendre conjecture that there should be at least one prime between n^2 and (n+1)^2.

Examples

			a(10) = 2 because between 10*(10+1)/2=55 and (10+1)*(10+2)/2=66 there are 2 primes: 59, 61.
		

Links

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

Crossrefs

A000217, A014085, A065383, A065384.

Essentially the same as A066888 and A090970.

Programs

Mathematica

Table[ PrimePi[n(n + 1)/2] - PrimePi[n(n - 1)/2], {n, 2, 96}]

Python

from sympy import primerange
def A065382(n): return sum(1 for p in primerange((n*(n+1)>>1)+1,((n+2)*(n+1)>>1)+1)) # Chai Wah Wu, May 22 2025

Extensions

Definition improved by Robert G. Wilson v, Apr 22 2003