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

3, 10, 18, 60, 80, 108, 360, 480, 648, 2160, 2880, 3888, 12960, 17280, 23328, 77760, 103680, 139968, 466560, 622080, 839808, 2799360, 3732480, 5038848, 16796160, 22394880, 30233088, 100776960, 134369280, 181398528, 604661760, 806215680, 1088391168

Offset: 1

Author

M. F. Hasler, Feb 09 2018

Keywords

• nonn
• base
• easy

Comments

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.

See A294090 for the base-10 variant, which is the main entry for this family of sequences, and A298977 for the base-7 variant.

Examples

			3 = (6-3), therefore 3 is in the sequence.
Denoting xyz[6] the base-6 expansion (i.e., x*6^2 + y*6 + z), we have:
10 = 14[6] = (6-1)*(6-4), therefore 10 is in the sequence.
18 = 30[6] = (6-3)*(6-0), therefore 18 is in the sequence.
80 = 212[6] = (6-2)*(6-1)*(6-2), therefore 80 is in the sequence.
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.
		

Links

• Colin Barker, Table of n, a(n) for n = 1..1000
• Index entries for linear recurrences with constant coefficients, signature (0,0,6).

Crossrefs

Cf. A294090, A298977.

Programs

Mathematica

LinearRecurrence[{0, 0, 6}, {3, 10, 18, 60, 80}, 50] (* Paolo Xausa, Jul 09 2025 *)

PARI

is(n,b=6)={n==prod(i=1,#n=digits(n,b),b-n[i])}

PARI

a(n)=if(n>5,a(n%3+3)*6^(n\3-1),[3,10,18,60,80][n])

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

Formula

a(n+3) = 6 a(n) for all n >= 2.

G.f.: x*(3 + 10*x + 18*x^2 + 42*x^3 + 20*x^4) / (1 - 6*x^3). - Colin Barker, Feb 09 2018