A298977 Base-7 complementary numbers: n equals the product of the 7 complement (7-d) of its base-7 digits d.

12, 84, 120, 588, 840, 4116, 5880, 28812, 41160, 201684, 288120, 1411788, 2016840, 9882516, 14117880, 69177612, 98825160, 484243284, 691776120, 3389702988, 4842432840, 23727920916, 33897029880, 166095446412, 237279209160, 1162668124884, 1660954464120

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 7 times a smaller term, are a(1) = 12 and a(3) = 120.

See A294090 for the base-10 variant, which is the main entry, and A298976 for the base-6 variant.

Examples

			Denoting xyz[7] the base-7 expansion (of n = x*7^2 + y*7 + z), we have:
12 = 15[7] = (7-1)*(7-5), therefore 12 is in the sequence.
84 = 150[7] = (7-1)*(7-5)*(7-0), therefore 84 is in the sequence.
120 = 231[7] = (7-2)*(7-3)*(7-1), therefore 120 is in the sequence.
Since the expansion of 7*x in base 7 is that of x with a 0 appended, if x is in the sequence, then 7*x = x*(7-0) 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, 7).

Crossrefs

Cf. A294090, A298976.

Programs

Mathematica

LinearRecurrence[{0, 7}, {12, 84, 120}, 30] (* Paolo Xausa, Jul 12 2025 *)

PARI

is(n,b=7)={n==prod(i=1,#n=digits(n,b),b-n[i])}

PARI

a(n)=[84,120][n%2+(n>1)]*7^(n\2-1)

PARI

Vec(12*x*(1 + 7*x + 3*x^2) / (1 - 7*x^2) + O(x^60)) \\ Colin Barker, Feb 10 2018

Formula

a(n+2) = 7 a(n) for all n >= 2.

From Colin Barker, Feb 10 2018: (Start)

G.f.: 12*x*(1 + 7*x + 3*x^2) / (1 - 7*x^2).

a(n) = 12*7^(n/2) for n>1 and even.

a(n) = 120*7^((n-3)/2) for n>1 and odd.

(End)

Extensions

More terms from Colin Barker, Feb 10 2018