A382963 Prime index gaps between consecutive full reptend primes.

3, 1, 1, 1, 5, 2, 1, 7, 4, 1, 2, 3, 4, 2, 1, 2, 4, 2, 1, 4, 1, 1, 8, 3, 5, 2, 1, 1, 4, 3, 5, 4, 1, 1, 1, 1, 3, 5, 1, 2, 6, 4, 2, 6, 1, 2, 3, 9, 1, 1, 5, 2, 4, 5, 1, 2, 2, 1, 1, 5, 1, 2, 3, 2, 1, 1, 1, 2, 1, 1, 5, 2, 1, 2, 3, 1, 1, 4, 5, 1, 1, 1, 4, 2, 2, 5, 1

Offset

1,1

Comments

This sequence gives the number of primes between consecutive full reptend primes, where a full reptend prime is a prime p for which 10 is a primitive root modulo p.

Links

Table of n, a(n) for n=1..87.

Formula

a(n) = pi(r(n+1)) - pi(r(n)), where r(n) is the n-th full reptend prime and pi(p) gives the prime index of p.

a(n) = A060257(n+1) - A060257(n).

Example

The full reptend primes begin 7 (index 4), 17 (index 7), 19 (index 8), 23 (index 9). Then:
a(1) = 7 - 4 = 3,
a(2) = 8 - 7 = 1,
a(3) = 9 - 8 = 1.

Prog

(Python)
from sympy import isprime, primerange, primepi
def is_full_reptend_prime(p):
  if not isprime(p): return False
  k, mod = 1, 10 % p
  while mod != 1:
    mod = (mod * 10) % p
    k += 1
    if k >= p: return False
  return k == p - 1
primes = list(primerange(2, 1000))
reptends = [p for p in primes if is_full_reptend_prime(p)]
gaps = [primepi(reptends[i+1]) - primepi(reptends[i]) for i in range(len(reptends)-1)]
print(gaps)
(Python)
from sympy import nextprime, n_order
def A382963_gen(): # generator of terms
    p, c = 7, 0
    while True:
        p, c = nextprime(p), c+1
        if n_order(10, p)==p-1:
            yield c
            c = 0
A382963_list = list(islice(A382963_gen(), 87)) # Chai Wah Wu, Apr 10 2025

Crossrefs

Partial differences of A060257.

Cf. A000040, A000720, A001913.

Sequence in context: A385198 A081060 A255811 * A131268 A109221 A369227

Adjacent sequences: A382960 A382961 A382962 * A382964 A382965 A382966

Keyword

nonn,easy

Author

Kyle Wyonch, Apr 10 2025

Status

approved