A247699 -id:A247699 - cp's OEIS frontend

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

James G. Merickel, Feb 05 2010

			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.

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

Cf. A247593, A247699.

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

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

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

A247593 Smallest prime not the middle of one 4 digits longer in base n.

Original entry on oeis.org

5, 1009, 764051, 7346914823

Offset: 2

James G. Merickel, Sep 20 2014

a(n), n > 6, is intractable, and a(6) requires extensive resources: There are 360 candidate numbers for any candidate prime, all of which need to be composite, prefixing 30 2-digit numbers and suffixing the 12 ending in either 1 or 5. This compares with 400 for base 5, but in the base-6 case divisibility by 2 and 3 are already ruled out.

			In base 2--binary, decimal 2 and 3 have representations 10 and 11; and binary 101001 and 101111 represent decimal 41 and 47, so that a(2) > 3.  Binary 101--decimal 5--has the 4 binary candidates 1010101, 1010111, 1110101, and 1110111--decimal 85, 87, 117 and 119--requiring consideration for primality, but all are composite: a(2)=5.

Cf. A172514, A247699.

ok(n,b)=my(D=b^#digits(n,b),b2=b^2);forstep(k=b^3*D+n*b2,b2*(b2-1)*D+n*b2,D*b2, if(nextprime(k)Charles R Greathouse IV, Sep 20 2014

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

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