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

1,1

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.

Links

Colin Barker, Table of n, a(n) for n = 1..1000

Index entries for linear recurrences with constant coefficients, signature (0, 7).

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)

Example

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.

Mathematica

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

Prog

(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

Crossrefs

Cf. A294090, A298976.

Sequence in context: A165127 A213347 A075476 * A392988 A213784 A085409

Adjacent sequences: A298974 A298975 A298976 * A298978 A298979 A298980

Keyword

nonn,base,easy

Author

M. F. Hasler, Feb 09 2018

Extensions

More terms from Colin Barker, Feb 10 2018

Status

approved