A236356 a(n) is the concatenation of the numbers k, 2 <= k <= 9, such that the base-k representation of n, read as a decimal number, is prime; a(n) = 0 if there is no such base.

0, 3456789, 2456789, 3, 246789, 5, 4689, 57, 68, 379, 48, 9, 45, 0, 68, 59, 47, 0, 468, 0, 59, 37, 245, 0, 68, 5, 6, 59, 47, 0, 78, 0, 568, 39, 8, 0, 469, 7, 689, 0, 5, 0, 4789, 0, 6, 3, 24, 9, 8, 7, 0, 7, 4, 0, 4689, 5, 8, 3, 78, 0, 49, 0, 5, 9, 8, 9, 368, 5

Offset: 1

Author

Vladimir Shevelev, Jan 23 2014

Keywords

• nonn
• base

Comments

Composite numbers n for which a(n)=0 we call absolute composite numbers.

Almost evidently that almost all integers are absolute composite numbers. Moreover, since the number of primes<=x containing no at least one digit is o(pi(x)), then, for almost all positions of prime n, a(n)=0. It is interesting to obtain an upper estimate of number of nonzero positions in the sequence, more exactly, than o(x/log(x)).

Only O(sqrt x) numbers up to x have nonzero values in this sequence. - Charles R Greathouse IV, Jan 24 2014

Examples

			Let n = 29. In bases 2, 3, ..., 9 the representations of 29 are 11101_2, 1002_3, 131_4, 104_5, 45_6, 41_7, 35_8, 32_9. In this list only 131_4 and 41_7 are primes, so a(29) = 47.
The sequence of numbers whose representations in bases 4 and 7, read in decimal, are primes are the numbers n such that a(n) contains the digits 4 and 7: 2, 3, 5, 17, 29, 43, ....
		

Links

• Peter J. C. Moses, Table of n, a(n) for n = 1..5000

Crossrefs

Cf. A052026.

Programs

Python

from sympy import isprime
from sympy.ntheory import digits
def c(n, b): return isprime(int("".join(map(str, digits(n, b)[1:]))))
def a(n): return int("0"+"".join(k for k in "23456789" if c(n, int(k))))
print([a(n) for n in range(1, 68)]) # Michael S. Branicky, Sep 22 2022

Extensions

Name clarified by Jon E. Schoenfield, Sep 21 2022