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Number of 6-ary sequences with primitive period n.
7

%I #22 Apr 14 2021 22:25:31

%S 1,6,30,210,1260,7770,46410,279930,1678320,10077480,60458370,

%T 362797050,2176734420,13060694010,78363884130,470184976590,

%U 2821108227840,16926659444730,101559946544280,609359740010490,3656158379595540,21936950640097710,131621703479470050

%N Number of 6-ary sequences with primitive period n.

%C Equivalently, output sequences with primitive period n from a simple cycling shift register.

%H Alois P. Heinz, <a href="/A054721/b054721.txt">Table of n, a(n) for n = 0..400</a>

%H E. N. Gilbert and J. Riordan, <a href="http://projecteuclid.org/euclid.ijm/1255631587">Symmetry types of periodic sequences</a>, Illinois J. Math., 5 (1961), 657-665.

%F a(n) = Sum mu(d)*6^(n/d); d|n.

%F a(0) = 1, a(n) = n * A032164(n).

%F G.f.: 1 + 6 * Sum_{k>=1} mu(k) * x^k / (1 - 6*x^k). - _Ilya Gutkovskiy_, Apr 14 2021

%p with(numtheory):

%p a:= n-> `if`(n=0, 1, add(mobius(d)*6^(n/d), d=divisors(n))):

%p seq(a(n), n=0..30); # _Alois P. Heinz_, Oct 21 2012

%t a[0] = 1; a[n_] := Sum[MoebiusMu[d]*6^(n/d), {d, Divisors[n]}]; Table[a[n], {n, 0, 30}] (* _Jean-François Alcover_, Mar 11 2014 *)

%Y Column k=6 of A143324.

%K nonn

%O 0,2

%A _N. J. A. Sloane_, Apr 20 2000