A362663 - OEIS

%I #25 Jun 16 2023 13:44:55

%S 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,

%T 7,9,10,10,10,13,14,13,16,15,14,14,17,17,15,17,17,16,16,18,18,20,22,

%U 18,19,19,18,19,17,19,25,27,27,30,31,37,35,35,34,34

%N 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].

%C 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.

%H Ya-Ping Lu, <a href="/A362663/a362663.pdf">A plot of a(n) for n up to 100000</a>

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

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

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

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

%e a(1) =        primepi(1^2+1) + primepi(1^2-1) - 2*primepi(1^2) =   1+0-2*0 = 1.

%e a(2) = a(1) + primepi(2^2+2) + primepi(2^2-2) - 2*primepi(2^2) = 1+3+1-2*2 = 1.

%e a(3) = a(2) + primepi(3^2+3) + primepi(3^2-3) - 2*primepi(3^2) = 1+5+3-2*4 = 1.

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

%o (Python)

%o from sympy import primerange; a0 = 0; L = []

%o def ct(m1, m2): return len(list(primerange(m1, m2)))

%o 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

%o print(*L, sep = ", ")

%o (PARI) a(n) = sum(i=1, n, primepi(i^2+i) + primepi(i^2-i) - 2*primepi(i^2)); \\ _Michel Marcus_, May 24 2023

%Y Cf. A014085, A089610, A094189, A192391.

%K sign

%O 1,4

%A _Ya-Ping Lu_, Apr 29 2023