A172514 First prime not the middle of a prime two digits longer in base n.

3, 7, 19, 97, 823, 3499, 2777, 6827, 2437, 21523, 300299, 446273, 339769, 1168523, 14117417, 29227421, 14160061, 78521987, 161187707, 1200085823, 2125209127, 1369430897, 56378083771, 26054006611, 76375900241, 290373503549, 640442460709

Offset: 2

Author

James G. Merickel, Feb 05 2010

Keywords

• nonn
• base
• more

Examples

			In base n=10, 2437 is the least prime such that all numbers of the form x2437y where x and y are digits [1..9] are composite, so a(10)=2437.
		

Crossrefs

Cf. A032734 (in base 10 and not limited to primes).

Cf. A247593, A247699.

Programs

PARI

isok(p, n) = my(m=logint(p,n)+1); for (x=1, n-1, my(q = x*n^m+p); for (y=1, n-1, if (isprime(n*q+y), return (0)););); return(1);
a(n) = my(p=2); while (!isok(p, n), p=nextprime(p+1)); p; \\ Michel Marcus, Sep 04 2022

Python

from sympy import isprime, nextprime
def digits(n, b):
    c = 0
    while n >= b: n //= b; c += 1
    return c + 1
def a(n):
    p = 2
    while True:
        d, p1, found = digits(p, n), n*p, True
        for f in range(n**(d+1), n**(d+2), n**(d+1)):
            for e in range(0, n, 2) if (f+p1)%2 else range(1, n, 2):
                if isprime(f + p1 + e): found = False; break
            if not found: break
        if found: return p
        p = nextprime(p)
print([a(n) for n in range(2, 15)]) # Michael S. Branicky, Sep 05 2022

Extensions

a(24)-a(26) added by James G. Merickel, Sep 22 2014

a(26) removed (see user talk page) by Bill McEachen, Sep 03 2022

a(26) from Michael S. Branicky, Sep 20 2022

a(27) from Michael S. Branicky, Jul 10 2023

a(28) from Michael S. Branicky, Jul 12 2023