A176559 -id:A176559 - cp's OEIS frontend

A309333 The number of primes between two consecutive lucky primes, bounds excluded.

1, 1, 4, 0, 1, 4, 1, 0, 8, 4, 1, 5, 2, 0, 4, 7, 1, 3, 2, 2, 6, 1, 1, 5, 2, 6, 5, 3, 1, 1, 0, 1, 4, 6, 1, 4, 1, 4, 9, 5, 7, 0, 0, 2, 2, 5, 1, 3, 0, 8, 4, 1, 5, 2, 18, 0, 9, 3, 1, 1, 9, 2, 4, 5, 3, 2, 6, 5, 4, 9, 3, 4, 11, 1, 1, 3, 4, 20, 0, 8, 2, 4, 3, 3, 15, 6

Offset: 1

Hauke Löffler, Jul 24 2019

nonn

			a(1): Between the first two lucky primes (3, 7) there is one prime (5).
a(3): Between 13 and 31 there are 4 primes (17, 19, 23, 29).

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A000959, A031157, A309334, A176559.

def count_primes_between(a, b):
  return len(prime_range(a+1, b))
[count_primes_between(A031157[i], A031157[i+1]) for i in range (len(A031157[0:20])-1)]

A307499 The number of primes between two consecutive prime Lucas numbers, bounds excluded.

Original entry on oeis.org

0, 1, 0, 4, 4, 30, 51, 230, 170, 657, 216347, 3009722, 16603784, 288244979, 4566061654, 192922096576, 20592039889787, 854140717540139, 7734073644382760578105

Offset: 1

Hauke Löffler, Jul 24 2019

			a(0): between the first two prime Lucas numbers (2,3) there are 0 primes.
a(3): between 11 and 29 there are 4 primes (13, 17, 19, 23).

Kim Walisch, Fast C++ prime counting function implementation (primecount).

Cf A005479, A134850, A176559.

Differences@ PrimePi@ Select[LucasL@ Range[0, 70], PrimeQ] - 1 (* Giovanni Resta, Jul 28 2019 *)

# uses[A005479]
def count_primes_between(a, b):
    return len(prime_range(a+1, b))
[count_primes_between(A005479[i], A005479[i+1]) for i in range(len(A005479)-1)]

a(14)-a(18) from Giovanni Resta, Jul 28 2019

a(19) using Kim Walisch's primecount, from Amiram Eldar, May 14 2023

A309321 The number of primes between two consecutive palindromic primes, bounds excluded.

Original entry on oeis.org

0, 0, 0, 0, 20, 5, 3, 5, 0, 21, 5, 2, 1, 52, 4, 3, 0, 17, 0, 1104, 21, 7, 73, 9, 105, 35, 8, 54, 51, 11, 34, 43, 78, 8, 52, 29, 19, 10, 80, 50, 22, 33, 78, 53, 9, 994, 11, 17, 26, 7, 20, 49, 75, 12, 109, 100, 27, 16, 12, 16, 32, 48, 28, 69, 32, 42, 6, 56, 48

Offset: 1

Hauke Löffler, Jul 23 2019

			a(0): Between the first two palindromic primes (2,3) there are 0 primes.
a(6): Between 101 and 131 there are 5 primes (103, 107, 109, 113, 127).

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Hauke Löffler)

Cf. A002385, A037010, A075807, A176559.

#Palindromic primes
def count_primes_between(a,b):
    return len(prime_range(a+1,b))
[count_primes_between(A002385[i],A002385[i+1]) for i in range (len(A002385)-1)]
# Alternative:
def A309321list(bound):
    L = []; p = 2
    while p < bound:
        p = next_prime(p)
        delta = 0
        while not Word(p.digits()).is_palindrome():
            delta += 1
            p = next_prime(p)
        L.append(delta)
    return L
A309321list(18181) # Peter Luschny, Jul 23 2019

a(n) = A075807(n+1) - A075807(n) - 1. - Jinyuan Wang, Jul 24 2019

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A309333 The number of primes between two consecutive lucky primes, bounds excluded.

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A307499 The number of primes between two consecutive prime Lucas numbers, bounds excluded.

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A309321 The number of primes between two consecutive palindromic primes, bounds excluded.

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