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Numbers k such that Phi(k, 12) is prime, where Phi is the cyclotomic polynomial.
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%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