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a(n) = phi(12^n-1), where phi is Euler's totient function ( A000010).
+10
13
10, 120, 1560, 13440, 226200, 2021760, 32518360, 274391040, 4534807680, 51953616000, 646094232960, 4662793175040, 97266341877120, 1070382142166400, 13666309113600000, 109897747141754880, 2016918439151095000, 17518491733377024000, 290436363064202660760
MATHEMATICA
EulerPhi[12^Range[30] - 1]
PROG
(PARI) {a(n) = eulerphi(12^n-1)}
CROSSREFS
phi(k^n-1): A053287 (k=2), A295500 (k=3), A295501 (k=4), A295502 (k=5), A366623 (k=6), A366635 (k=7), A366654 (k=8), A366663 (k=9), A295503 (k=10), A366685 (k=11), this sequence (k=12).
Smallest prime dividing 12^n + 1.
+10
10
2, 13, 5, 7, 89, 13, 5, 13, 17, 7, 5, 13, 89, 13, 5, 7, 153953, 13, 5, 13, 41, 7, 5, 13, 17, 13, 5, 7, 89, 13, 5, 13, 769, 7, 5, 13, 89, 13, 5, 7, 17, 13, 5, 13, 89, 7, 5, 13, 7489, 13, 5, 7, 89, 13, 5, 13, 17, 7, 5, 13, 41, 13, 5, 7, 36097, 13, 5, 13, 89, 7
Largest prime factor of 12^n+1.
+10
7
2, 13, 29, 19, 233, 19141, 20593, 13063, 260753, 1801, 85403261, 57154490053, 2227777, 222379, 13156924369, 35671, 1200913648289, 66900193189411, 122138321401, 905265296671, 67657441, 1885339, 68368660537, 49489630860836437, 592734049, 438472201
MATHEMATICA
Table[FactorInteger[12^n + 1][[-1, 1]], {n, 0, 20}]
CROSSREFS
Cf. A178248, A006530, A002587, A074476, A274903, A074478, A274904, A227575, A274905, A002592, A003021, A062308, A002590, A366712, A366713, A366714, A366715, A366716, A366717, A366718, A366719, A324941.
Largest prime factor of 12^n - 1.
+10
4
11, 13, 157, 29, 22621, 157, 4943, 233, 80749, 22621, 266981089, 20593, 20369233, 13063, 22621, 260753, 74876782031, 80749, 29043636306420266077, 85403261, 8177824843189, 57154490053, 321218438243, 2227777, 12629757106815551, 20369233, 86769286104133
LINKS
J. Brillhart et al., Factorizations of b^n +- 1, Contemporary Mathematics, Vol. 22, Amer. Math. Soc., Providence, RI, 3rd edition, 2002.
MATHEMATICA
Table[FactorInteger[12^n - 1][[-1, 1]], {n, 40}]
PROG
(Magma) [Maximum(PrimeDivisors(12^n-1)): n in [1..40]];
CROSSREFS
Cf. A024140, A006530, A005420, A074477, A274906, A074479, A274907, A074249, A274908, A274909, A005422, A274910, A366707, A366708, A366709, A366710, A366711, A366717, A366720.
a(n) = period length of the sequence A020639(n^k - 1), k >= 1.
+10
1
1, 1, 1, 1, 1, 2, 1, 1, 1, 12, 1, 10, 1, 1, 1, 60, 1, 10, 1, 1, 1, 18, 1, 2, 1, 1, 1, 660, 1, 66, 1, 1, 1, 1, 1, 10, 1, 1, 1, 4620, 1, 6, 1, 1, 1, 660, 1, 2, 1, 1, 1, 31878, 1, 2, 1, 1, 1, 197340, 1, 5742, 1, 1, 1, 1, 1, 52026, 1, 1, 1, 440220, 1, 28014, 1, 1, 1, 4, 1, 2610, 1, 1, 1, 28014, 1, 2, 1, 1, 1, 3693690, 1, 2, 1, 1, 1, 1, 1, 7590, 1, 1, 1, 1642460820
COMMENTS
For n = 2, the sequence A020639(n^k - 1) is not periodic (see A049479), but it is such for any n >= 3.
FORMULA
For odd n >= 3, a(n) = 1.
EXAMPLE
a(8) = 2 is the period length of A010705. a(12) = 12 is the period length of A366717.
PROG
(PARI) { a368811(n) = my(r=[], z); forprime(p=2, factor(n-1)[1, 1], if(n%p==0, next); z=znorder(Mod(n, p)); if(!#r || vecmin(apply(x->z%x, r)), r=concat(r, [z])) ); lcm(r); }
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