A386873 Smallest prime p such that (p-k-1)/k is also prime for all k=1..n.

5, 7, 13, 13, 2129401, 62198641, 62198641, 62198641, 28641897012961, 5193520377811921, 64090546658938801

Offset: 1

Author

Marc Morgenegg, Aug 06 2025

Keywords

• nonn
• more
• new

Comments

All terms are the larger member of a twin prime pair, since for k=1 (p-1-1)/1=p-2 must be prime.

Examples

			a(4) = 13 is the smallest prime p such that (p-k-1)/k is prime for all k=1..4. (13-2)/1=11, (13-3)/2=5, (13-4)/3=3, (13-5)/4=2.
		

Links

• Marc Morgenegg, Python Program

Crossrefs

Subsequence of A006512.

Cf. A089531.

Programs

Python

from itertools import islice
from gmpy2 import is_prime, next_prime
def agen(): # generator of terms
    n = p = 1
    while True:
        k, p = 1, next_prime(p); q, r = p - 2, 0
        while r == 0 and is_prime(q): k += 1; q, r = divmod(p - k - 1, k)
        while n < k: n += 1; yield int(p)
print(list(islice(agen(), 8))) # Michael S. Branicky, Aug 14 2025

Python

from itertools import count
from math import lcm
from sympy import isprime
def A386873(n):
    m = lcm(*range(1,n+1))
    for i in count(m+1,m):
        if isprime(i) and all(isprime((i-1)//j-1) for j in range(1,n+1)):
            return i # Chai Wah Wu, Aug 20 2025

Formula

a(n) == 1 mod lcm(1,2,...,n). - Chai Wah Wu, Aug 20 2025

Extensions

a(9)-a(11) from Jinyuan Wang, Aug 15 2025