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A227864
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Smallest base in which n's digital reversal is prime, or 0 if no such base exists.
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2
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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
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OFFSET
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0,3
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COMMENTS
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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.
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LINKS
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EXAMPLE
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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.
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PROG
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(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
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CROSSREFS
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KEYWORD
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nonn,base,easy
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AUTHOR
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STATUS
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approved
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