A294090 Base-10 complementary numbers: n equals the product of the 10's complement of its digits.

5, 18, 35, 50, 180, 315, 350, 500, 1800, 3150, 3500, 5000, 18000, 31500, 35000, 50000, 180000, 315000, 350000, 500000, 1800000, 3150000, 3500000, 5000000, 18000000, 31500000, 35000000, 50000000, 180000000, 315000000, 350000000, 500000000, 1800000000

Offset

1,1

Comments

The only primitive terms of the sequence, i.e., not equal to 10 times a smaller term, are 5, 18, 35 and 315.

For base 2, 3, 4 and 5, the corresponding sequences are less interesting: b = 2 yields powers of 2, A000079; b = 3 yields 4 times powers of 3, A003946 \ {1}; b = 4 yields {2, 6}*{4^k, k>=0} = A122756 = 2*A084221; b = 5 yields 8*{5^k, k>=0} = A128625 \ {1}.

See A298976 for base-6 complementary numbers. Base 7 yields {12, 120}*{7^k, k>=0}, cf. A298977. The linked web page (in French) gives also examples for base-100 complementary numbers, e.g., 198 = (100 - 1)*(100 - 98), 1680 = (100 - 16)*(100 - 80), ..., and for base-1000 complementary numbers.

Links

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

G. Villemin, Nombres complémentés (in French).

Index entries for linear recurrences with constant coefficients, signature (0,0,0,10).

Formula

a(n+4) = 10 a(n) for all n >= 3.

G.f.: x*(5 + 18*x + 35*x^2 + 50*x^3 + 130*x^4 + 135*x^5) / (1 - 10*x^4). - Colin Barker, Feb 09 2018

Example

5 = (10-5), therefore 5 is in the sequence.
18 = (10-1)*(10-8), therefore 18 is in the sequence.
35 = (10-3)*(10-5), therefore 35 is in the sequence.
315 = (10-3)*(10-1)*(10-5), therefore 315 is in the sequence.
If x is in the sequence, then 10*x = concat(x,0) = x*(10-0) is in the sequence.

Mathematica

LinearRecurrence[{0, 0, 0, 10}, {5, 18, 35, 50, 180, 315}, 40] (* Harvey P. Dale, Mar 02 2024 *)

Prog

(PARI) is(n, b=10)={n==prod(i=1, #n=digits(n, b), b-n[i])}
(PARI) a(n)=if(n>6, a((n-3)%4+3)*10^((n-3)\4), [5, 18, 35, 50, 180, 315][n])
(PARI) Vec(x*(5 + 18*x + 35*x^2 + 50*x^3 + 130*x^4 + 135*x^5) / (1 - 10*x^4) + O(x^60)) \\ Colin Barker, Feb 09 2018

Crossrefs

Cf. A298976, A298977.

Sequence in context: A031004 A063120 A031051 * A038346 A220243 A065007

Adjacent sequences: A294087 A294088 A294089 * A294091 A294092 A294093

Keyword

nonn,base,easy

Author

M. F. Hasler, Feb 09 2018

Status

approved