OFFSET
1,1
COMMENTS
Base-16 (a.k.a. hexadecimal, sexadecimal, senidenary or hexadecadic) pandigital primes must have at least 17 hexadecimal digits (i.e. they are larger than 16^16 = 2^64 > 10^19), since sum(d_i 16^i) = sum(d_i) (mod 15), and 0+1+...+14+15 is divisible by 15. So the smallest ones should be of the form "101234567...." in base 16, where "...." is a permutation of "89ABCDEF".
The same reasoning shows that numbers of this form ("1012...") are congruent to 1 modulo 15 and thus modulo 30 (since also = 1 [mod 2]). This explains that all terms < 2*16^16 end in the (decimal!) digit 1.
a(n) == 1 (mod 30) for a(n) < 2^65 = 3.69*10^19.
LINKS
Alonso Del Arte, Classifications of prime numbers - By representation in specific bases, OEIS Wiki as of Mar 19 2010.
M. F. Hasler, Reply to A. Del Arte's post "Pandigital primes in bases 8,12,..." on the SeqFan list, Mar 19 2010.
Wikipedia, Hexadecimal: Etymology; Pandigital.
PROG
(PARI) pdp( b=16/*base*/, c=99/* # of terms to produce */) = { my(t, a=[], bp=vector(b, i, b^(b-i))~, offset=b*(b^b-1)/(b-1)); for( i=1, b-1, offset+=b^b; for( j=0, b!-1, isprime(t=offset-numtoperm(b, j)*bp) | next; #(a=concat(a, t))<c | return(vecsort(a))))} /* NOTE: Due to the implementation of numtoperm, the returned list may be incomplete towards its end. Thus computation of more than the required # of terms is recommended. [The initial digits of the base-16 expansion of the terms allow one to know up to where it is complete.] You may use a construct of the form: vecextract(pdp(16, 999), "1..20")) */
CROSSREFS
KEYWORD
nonn,base
AUTHOR
M. F. Hasler, May 27 2010
EXTENSIONS
Edited by Charles R Greathouse IV, Aug 02 2010
STATUS
approved