A175273 Base-16 pandigital primes.

18528729602926047181, 18528729602926100221, 18528729602926108411, 18528729602926112701, 18528729602926116331, 18528729602926234591, 18528729602926235071, 18528729602927029471, 18528729602927225551

Offset: 1

Author

M. F. Hasler, May 27 2010

Keywords

• nonn
• base

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.

Crossrefs

Cf. A138837, A050288, A175271 - A175280.

Programs

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))
				

Extensions

Edited by Charles R Greathouse IV, Aug 02 2010