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

1,2

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

Links

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

Example

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, ....

Prog

(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

Crossrefs

Cf. A052026.

Sequence in context: A258517 A258510 A254997 * A082247 A036473 A154871

Adjacent sequences: A236353 A236354 A236355 * A236357 A236358 A236359

Keyword

nonn,base

Author

Vladimir Shevelev, Jan 23 2014

Extensions

Name clarified by Jon E. Schoenfield, Sep 21 2022

Status

approved