A362663 a(n) is the partial sum of b(n), which is defined to be the difference between the numbers of primes in (n^2, n^2 + n] and in (n^2 - n, n^2].

1, 1, 1, 2, 2, 3, 2, 2, 2, 5, 6, 6, 6, 6, 8, 10, 8, 6, 5, 5, 5, 6, 5, 5, 4, 4, 5, 5, 4, 4, 5, 5, 7, 7, 7, 9, 10, 10, 10, 13, 14, 13, 16, 15, 14, 14, 17, 17, 15, 17, 17, 16, 16, 18, 18, 20, 22, 18, 19, 19, 18, 19, 17, 19, 25, 27, 27, 30, 31, 37, 35, 35, 34, 34

Offset: 1

Author

Ya-Ping Lu, Apr 29 2023

Keywords

• sign

Comments

A plot of a(n) for n up to 100000 is given in Links. First negative term is a(177) = -7 and first zero term appears at n = 198.

Examples

			a(1) =        primepi(1^2+1) + primepi(1^2-1) - 2*primepi(1^2) =   1+0-2*0 = 1.
a(2) = a(1) + primepi(2^2+2) + primepi(2^2-2) - 2*primepi(2^2) = 1+3+1-2*2 = 1.
a(3) = a(2) + primepi(3^2+3) + primepi(3^2-3) - 2*primepi(3^2) = 1+5+3-2*4 = 1.
a(4) = a(3) + primepi(4^2+4) + primepi(4^2-4) - 2*primepi(4^2) = 1+8+5-2*6 = 2.
		

Links

• Ya-Ping Lu, A plot of a(n) for n up to 100000

Crossrefs

Cf. A014085, A089610, A094189, A192391.

Programs

PARI

a(n) = sum(i=1, n, primepi(i^2+i) + primepi(i^2-i) - 2*primepi(i^2)); \\ Michel Marcus, May 24 2023

Python

from sympy import primerange; a0 = 0; L = []
def ct(m1, m2): return len(list(primerange(m1, m2)))
for n in range(1,75): s = n*n; a = a0+ct(s,s+n+1)-ct(s-n+1,s); L.append(a); a0 = a
print(*L, sep = ", ")

Formula

a(n) = a(n-1) + primepi(n^2+n) + primepi(n^2-n) - 2*primepi(n^2).

a(n) = Sum_{i=1..n} (primepi(i^2+i) + primepi(i^2-i) - 2*primepi(i^2)).

a(n) = 2 + Sum_{i=2..n} (A089610(i) - A094189(i)), for n >= 2.

a(A192391(m)) = a(A192391(m)-1), for m >= 2.