login
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.
13
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
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
KEYWORD
nonn,base
AUTHOR
Vladimir Shevelev, Jan 23 2014
EXTENSIONS
Name clarified by Jon E. Schoenfield, Sep 21 2022
STATUS
approved