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A298976 Base-6 complementary numbers: n equals the product of the 6 complement (6-d) of its base-6 digits d.

%I A298976 #19 Jul 09 2025 10:16:32
%S A298976 3,10,18,60,80,108,360,480,648,2160,2880,3888,12960,17280,23328,77760,
%T A298976 103680,139968,466560,622080,839808,2799360,3732480,5038848,16796160,
%U A298976 22394880,30233088,100776960,134369280,181398528,604661760,806215680,1088391168
%N A298976 Base-6 complementary numbers: n equals the product of the 6 complement (6-d) of its base-6 digits d.
%C A298976 The only primitive terms of the sequence, i.e., not equal to 6 times a smaller term, are a(1) = 3, a(2) = 10 and a(5) = 80.
%C A298976 See A294090 for the base-10 variant, which is the main entry for this family of sequences, and A298977 for the base-7 variant.
%H A298976 Colin Barker, <a href="/A298976/b298976.txt">Table of n, a(n) for n = 1..1000</a>
%H A298976 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (0,0,6).
%F A298976 a(n+3) = 6 a(n) for all n >= 2.
%F A298976 G.f.: x*(3 + 10*x + 18*x^2 + 42*x^3 + 20*x^4) / (1 - 6*x^3). - _Colin Barker_, Feb 09 2018
%e A298976 3 = (6-3), therefore 3 is in the sequence.
%e A298976 Denoting xyz[6] the base-6 expansion (i.e., x*6^2 + y*6 + z), we have:
%e A298976 10 = 14[6] = (6-1)*(6-4), therefore 10 is in the sequence.
%e A298976 18 = 30[6] = (6-3)*(6-0), therefore 18 is in the sequence.
%e A298976 80 = 212[6] = (6-2)*(6-1)*(6-2), therefore 80 is in the sequence.
%e A298976 Since the expansion of 6*x in base 6 is that of x with a 0 appended, if x is in the sequence, then 6*x = x*(6-0) is in the sequence.
%t A298976 LinearRecurrence[{0, 0, 6}, {3, 10, 18, 60, 80}, 50] (* _Paolo Xausa_, Jul 09 2025 *)
%o A298976 (PARI) is(n,b=6)={n==prod(i=1,#n=digits(n,b),b-n[i])}
%o A298976 (PARI) a(n)=if(n>5,a(n%3+3)*6^(n\3-1),[3,10,18,60,80][n])
%o A298976 (PARI) Vec(x*(3 + 10*x + 18*x^2 + 42*x^3 + 20*x^4) / (1 - 6*x^3) + O(x^60)) \\ _Colin Barker_, Feb 09 2018
%Y A298976 Cf. A294090, A298977.
%K A298976 nonn,base,easy
%O A298976 1,1
%A A298976 _M. F. Hasler_, Feb 09 2018