User talk:M. F. Hasler/drafts/Pandigital primes

References :

Alonso Del Arte, <a href="http://oeis.org/wiki/Classifications_of_prime_numbers#By_representation_in_specific_bases">Classifications of prime numbers - By representation in specific bases</a>, OEIS Wiki as of Mar 19 2010.

M. F. Hasler, <a href="http://list.seqfan.eu/pipermail/seqfan/2010-March/004101.html">Reply to A. Del Arte's post "Pandigital primes in bases 8,12,..." on the SeqFan list</a>, Mar 19 2010.

Base-8, 12, 16, 20 & 36 pandigital primes will be A175271 - A175275:

*** NOTE ! minor problem -- the numbers below seem to be in disorder. ***

%I A175272
%S A175272 8989787252711,8989787311891,8989787313343,8989787458763,8989787707627,
%T A175272 8989787709211,8989787710927,8989788452371,8989787959879,8989787764211,
%U A175272 8989788261983,8989787806099,8989787992747,8989788241699,8989788262423
%N A175272 Base-12 pandigital primes. 

%C A175272 These numbers need to have at least 13 digits in base 12 since any permutation of the digits 0,...,9,A,B will result in a number divisible by 11. For the same reason, it must be digit different from 0 which is repeated. Thus the smallest terms in this sequence are written "10123456....." in base 12, where ..... is a permutation of {7,8,9,A,B}.

%H A175272 Alonso Del Arte, <a href="http://oeis.org/wiki/Classifications_of_prime_numbers#By_representation_in_specific_bases">Classifications of prime numbers - By representation in specific bases</a>, OEIS Wiki as of Mar 19 2010.

%H A175272 M. F. Hasler, <a href="http://list.seqfan.eu/pipermail/seqfan/2010-March/004101.html">Reply to A. Del Arte's post "Pandigital primes in bases 8,12,..." on the SeqFan list</a>, Mar 19 2010.

%e A175272 8989787252711, 8989787311891, 8989787313343, 8989787458763, ... are written "101234568A79B", "10123456B8A97", "10123456B98A7", "1012345769A8B", ... in base 12 (where A=digit 10, B=digit 11).

%o A175272 (PARI) pdp( b=12/* base */, c=20 /* #terms to print */)={ my(t,bp=vector(b,i,b^(b-i))~, offset=b*(b^b-1)/(b-1) /* to fix order of permutations CBA..321 => 012...9AB */); for( i=1,b-1, /* add initial digit */ offset += b^b; for( j=0,b!-1, isprime(t=offset-numtoperm(b,j)*bp) & !print1(t", ") & !c-- & return))}

%Y A175272 Cf. A138837, A050288, A175271, A175273, A175274, A175275.

%K A175272 nonn

%O A175272 1,1

%A A175272 M. F. Hasler (MHasler(AT)univ-ag.fr), Mar 19 2010

The numbers 17119607, 17120573, 17121077, 17135413, 17136869, 17127839, 17136029, 17132347, 17128931, 17148349, 17213239, 17245999, 17246951, 17247973, 17181683, 17213939, 17247203, 17159479, 17184119, 17200373, 17196383, 17253727, 17253853, 17286557, 17257759, 17265949, 17185463, 17185981, 17196733, 17229479, 17229983, 17194171, 17202347, 17287859, 17235107, 17164757, 17202389, 17258711, 17223571, 17288083, 17292563, 17348983, 17349949, 17349991, 17352623, 17352637, 17360701, 17353967, 17365237, 17361247, 17361373,... seem to be base-8 pandigital primes.

These are maybe not the smallest, I created them by appending a permutation of digits 0..7 to a given (most significant) digit (0 => no prime, since these numbers are divisible by 7; 1 => yields the above primes, 2=> ?,...)

In base 8 they read "101234567", "101236475", "101237465", "101273465", "101276345", "101254637", "101274635", "101265473", "101256743", "101324675", "101523467", "101623457", "101625347",...

PS: cryptic PARI code: {c=0; p8=vector(8,i,8^(8-i))~; forstep(j=0,7,1, offset=8*(8^8-1)/7+j*8^8; for(i=0,8!-1, isprime(t=offset-numtoperm(8,i)*p8) & !print1(t", ") & c++>50 & return))}

Base 12 pandigital primes seem to start with 8989787252711, 8989787311891, 8989787313343, 8989787458763, 8989787707627, 8989787709211, 8989787710927, 8989788452371, 8989787959879, 8989787764211, 8989788261983, 8989787806099, 8989787992747, 8989788241699, 8989788262423, 8989787974883, 8989787810719, 8989788495007, 8989787999743, 8989788058351, ...

= "101234568A79B", "10123456B8A97", "10123456B98A7", "1012345769A8B", "1012345869AB7", "101234586A9B7", "101234586B9A7", "1012345B68A97"...

Hexa: 18528729602926047181, 18528729602926100221, 18528729602926234591, 18528729602926112701, 18528729602926235071, 18528729602926108411, 18528729602926116331, 18528729602927029471, 18528729602930170831, 18528729602928082621, 18528729602930167741, 18528729602928082411,...

= "10123456789ABEFCD", "10123456789ACBEFD", "10123456789AECBDF", "10123456789ACEFBD", "10123456789AECDBF", "10123456789ACDEFB"....

base-20: 105148064265927977839670339, 105148064265927977839990337, 105148064265927977839838717, 105148064265927977848790339, 105148064265927977843159537, 105148064265927977846038379, 105148064265927977852278397, 105148064265927977848933979, 105148064265927977852157937, ...

= "10123456789ABCDEHIFGJ", "10123456789ABCDEJIFGH", "10123456789ABCDEIJGFH", "10123456789ABCDHEIFGJ",...

base-36: 106474205747327721970821813283682888755465951838540182351, 106474205747327721970821813283682888755465951838655934631, 106474205747327721970821813283682888755465951838716447391, 106474205747327721970821813283682888755465951838776957771, 106474205747327721970821813283682888755465951838718031211, 106474205747327721970821813283682888755465951838781855251, ...

= "10123456789ABCDEFGHIJKLMNOPQRSTUXYZWV", "10123456789ABCDEFGHIJKLMNOPQRSTWUVYXZ", "10123456789ABCDEFGHIJKLMNOPQRSTXUWYVZ", "10123456789ABCDEFGHIJKLMNOPQRSTYUXWZV", ...