A065384 -id:A065384 - cp's OEIS frontend

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

Reinhard Zumkeller, Nov 05 2001

nonn

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

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

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

A000217, A014085, A065383, A065384.

Essentially the same as A066888 and A090970.

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

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

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

A375752 a(n) is the difference between T=n*(n+1)/2 and the largest prime not exceeding T.

Original entry on oeis.org

0, 1, 3, 2, 2, 5, 5, 2, 2, 5, 5, 2, 2, 7, 5, 2, 4, 9, 11, 2, 2, 5, 7, 8, 2, 5, 5, 2, 2, 5, 5, 4, 2, 11, 5, 2, 2, 7, 9, 2, 16, 5, 7, 2, 12, 5, 5, 2, 16, 5, 5, 2, 2, 9, 13, 16, 2, 11, 7, 2, 2, 5, 11, 2, 4, 5, 5, 4, 8, 5, 7, 2, 8, 7, 9, 2, 2, 23, 11, 2, 12, 17, 11, 12, 2

Offset: 2

Hugo Pfoertner, Aug 26 2024

Cf. A000217, A007917, A065384, A075402, A375753, A375754.

Table[T=n(n+1)/2;T-If[PrimeQ[T],T,NextPrime[T,-1]],{n,2,86}] (* James C. McMahon, Sep 27 2024 *)

a(n) = my(t=n*(n+1)/2); t - precprime(t)

from sympy import prevprime
def A375752(n): return (t:=n*(n+1)//2) - prevprime(t) # Karl-Heinz Hofmann, Aug 27 2024

a(n) = A000217(n) - A065384(n). - Michel Marcus, Aug 27 2024

cp's OEIS Frontend

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

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