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A227864 Smallest base in which n's digital reversal is prime, or 0 if no such base exists. 2
0, 0, 3, 2, 0, 2, 2, 2, 4, 6, 2, 2, 2, 2, 2, 3, 8, 2, 3, 3, 2, 3, 2, 2, 2, 2, 2, 9, 2, 2, 6, 2, 4, 3, 2, 3, 12, 2, 6, 3, 2, 2, 6, 2, 2, 3, 2, 2, 2, 3, 2, 9, 2, 2, 3, 2, 2, 3, 2, 4, 12, 2, 2, 3, 12, 3, 6, 2, 2, 3, 10, 2, 6, 2, 2, 3, 10, 2, 26, 3, 2, 27, 2, 2 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

0 and 1 are not prime and are single digits in all bases, so no reversal of digits can make them prime. a(n) is therefore 0 for both.

4 is not prime and so cannot be prime if reversed in any base where it is a single digit. This leaves bases 2 and 3 where, upon reversal, it is 1 and 4 respectively. Neither are prime, and so a(4) is also 0.

Conjecture 1: 0, 1 and 4 are the only values where there is no base in which a digital reversal makes a prime.

It is clear that for any prime p, a(p) cannot be zero, since a(p)=p+1 is a solution if there is none smaller.

Conjecture 2: n = 2 is the only prime p which must be represented in base p+1, i.e. trivially, as a single digit, in order for its reversal to be prime.

Corollary: Since a(n) cannot be n itself -- reversing n in base n obtains 1, which is not prime -- this would mean that for all positive n except 2, a(n) < n.

Other than its small magnitude, a(n) = 2 occurs often due to the fact that a reversed positive binary number is guaranteed to be odd and thus stands a greater chance of being prime.

Similarly, many solutions exist solely because reversal removes all powers of the base from n, reducing the number of divisors. Thus based solely on observation:

Conjecture 3: With the restriction gcd(base,n) = 1, a(n) = 0 except for n = 2, 3 and 6k+-1, for positive integer k, i.e., terms of A038179.

LINKS

Carl R. White, Table of n, a(n) for n = 0..9999

EXAMPLE

9 in base 2 is 1001, which when reversed is the same and so not prime. In base 3 it is 100, which becomes 1 when reversed and also not prime. Base 4: 21 -> 12 (6 decimal), not prime; Base 5: 14 -> 41 (21 decimal), not prime; Base 6: 13 -> 31 (19 decimal), which is prime, so a(9) = 6, i.e., 6 is the smallest base in which 9's digital reversal is a prime number.

PROG

(Python)

from sympy import isprime

from sympy.ntheory.digits import digits

def okb(n, b):

    return isprime(sum(d*b**i for i, d in enumerate(digits(n, b)[1:])))

def a(n):

    for b in range(2, n+2):

        if okb(n, b): return b

    return 0

print([a(n) for n in range(84)]) # Michael S. Branicky, Sep 06 2021

CROSSREFS

Cf. A114018, A006567, A095179.

Positions of 2's: A204232.

Sequence in context: A054503 A281451 A246863 * A236603 A129576 A122861

Adjacent sequences:  A227861 A227862 A227863 * A227865 A227866 A227867

KEYWORD

nonn,base,easy

AUTHOR

Carl R. White, Nov 01 2013

STATUS

approved

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Last modified August 10 11:57 EDT 2022. Contains 356039 sequences. (Running on oeis4.)