%I #42 Mar 26 2021 08:38:50
%S 1,2,3,5,10,12,19,21,22,56,60,63,70,80,84,92,97,109,111,123,164,189,
%T 218,276,317,353,364,386,405,456,511,636,675,701,793,945,1090,1268,
%U 1272,1971,2088,2368,2482,2893,2966,3290,4161,4320,4533,4744,6357,7023,7430,7737,9499,9739
%N Numbers k such that Phi(k, 12) is prime, where Phi is the cyclotomic polynomial.
%C Numbers k such that A019330(k) is prime.
%C With some exceptions, terms of sequence are such that 12^n - 1 has only one primitive prime factor. 20 is an instance of such an exception, since 12^20 - 1 has a single primitive prime factor, 85403261, but Phi(20, 12) is divisible by 5, it is not prime.
%C a(n) is a duodecimal unique period length.
%e n Phi(n, 12)
%e 1 11
%e 2 13
%e 3 157
%e 4 5 * 29
%e 5 22621
%e 6 7 * 19
%e 7 659 * 4943
%e 8 89 * 233
%e 9 37 * 80749
%e 10 19141
%e 11 11 * 23 * 266981089
%e 12 20593
%e etc.
%t Select[Range[1728], PrimeQ[Cyclotomic[#, 12]] &]
%o (PARI) for( i=1, 1728, ispseudoprime( polcyclo(i, 12)) && print1( i", "))
%Y Cf. A019330, A072226, A138919-A138940.
%K nonn
%O 1,2
%A _Eric Chen_, Dec 16 2014
%E More terms from _Michel Marcus_, Dec 18 2014
%E More terms from _Amiram Eldar_, Mar 26 2021