A371512 a(n) is the smallest prime such that its representation in base n contains each of the digits 1,...,n-2 at least once and does not contain the digit 0 nor the digit n-1.

13, 37, 163, 1861, 22481, 304949, 5455573, 112345687, 2831681057, 68057976031, 1953952652167, 61390449569437, 2224884906436873, 77181689614101181, 3052505832274232281, 129003238915759600789, 6090208982148446231753, 276667213296398892309917, 13944042713948404997174231

Offset: 3

Author

Chai Wah Wu, Apr 10 2024

Keywords

• nonn
• base

Comments

Conjecture: for n>3, a(n) has digit sum 2+(n-2)(n-1)/2 if n is of the form 4k+3 and has digit sum 1+(n-2)(n-1)/2 otherwise.

Examples

			The corresponding base-n representations are:
n   a(n) in base n
------------------------
3   111
4   211
5   1123
6   12341
7   122354
8   1123465
9   11234567
10  112345687
11  1223456987
12  1123458a967
13  112345678ba9
14  11234567a8bc9
15  122345678acb9d
16  1123456789ceabd
		

Links

• Chai Wah Wu, Table of n, a(n) for n = 3..388
• Chai Wah Wu, Pandigital and penholodigital numbers, arXiv:2403.20304 [math.GM], 2024.

Crossrefs

Cf. A185122, A371194.

Programs

Python

from math import gcd
from sympy import nextprime
from sympy.ntheory import digits
def A371512(n):
    m, j = 1, 0
    if n > 3:
        for j in range(1,n-1):
            if gcd((n*(n-1)>>1)+j,n-1) == 1:
                 break
    if j == 0:
        for i in range(2,n-1):
            m = n*m+i
    elif j == 1:
        for i in range(1,n-1):
            m = n*m+i
    else:
        for i in range(2,1+j):
            m = n*m+i
        for i in range(j,n-1):
            m = n*m+i
    m -= 1
    while True:
        s = digits(m:=nextprime(m), n)[1:]
        if (not (0 in s or n-1 in s)) and len(set(s))==n-2:
            return m

Formula

For n>=3, a(n) >= (n^(n-1)-n)/(n-1)^2 + n^(n-2). If n = 4k+3 for k>0, then a(n) >= (n^(n-1)-n)/(n-1)^2 + n^(n-2) + n^(n-3) .