A140959 Number of distinct-digit primes in base n.

0, 1, 6, 6, 31, 130, 632, 4418, 34401, 283086, 2586883, 28637741, 336810311

Offset: 1

Author

Robert G. Wilson v, Jul 25 2008

Keywords

• nonn
• base
• hard
• more

Examples

			a(1) = 0; a(2) = 1 since only the prime 2 in base 2 has distinct integers, 10_2;
a(3) = 6 since the primes {2, 3, 5, 7, 11 & 19} in base 3 have distinct integers, {2_3, 10_3, 12_3, 21_3, 102_3, 201_3}; etc.
a(10) = 283086 because it is the partial sum of A073532.
		

Crossrefs

Cf. A073532.

Programs

Mathematica

f[b_] := Block[{c = 0, k = 1, lmt = b^b}, While[p = Prime@ k; p < lmt, k++; If[ Union[ Length /@ Split@ Sort@ IntegerDigits[p, b]] == {1}, c++ ]]; c]; Array[f, 6]

Python

from sympy import isprime
from itertools import permutations
def a(n):
    digs = "".join(str(i) for i in range(min(10, n)))
    if n > 10: digs += "".join(chr(ord("A")+i) for i in range(n-10))
    return sum(1 for i in range(1, n+1) for p in permutations(digs, i) if p[0] != '0' and isprime(int("".join(p), n)) )
print([a(n) for n in range(1, 10)]) # Michael S. Branicky, Dec 25 2021

Sage

def a(n):
    return sum(len(p.digits(n)) == len(set(p.digits(n))) for p in prime_range(n^n)) # Eric M. Schmidt, Oct 26 2014

Extensions

a(11)-a(12) from Eric M. Schmidt, Oct 29 2014

a(13) from Michael S. Branicky, Dec 25 2021