A065759 For a number k of length L, let f(k) be the sum of the products of the first i digits of k multiplied by the last L-i digits, for i from 1 to L-1, e.g., f(1234) = 1*234 + 12*34 + 123*4 = 1134. Sequence gives k such that f(k) = k.

0, 655, 1461, 1642, 2361, 3442, 6550, 14610, 16420, 23610, 34420, 65500, 146100, 164200, 236100, 344200, 655000, 1461000, 1642000, 2361000, 3442000, 6550000, 14610000, 16420000, 23610000, 34420000, 65500000, 146100000, 164200000, 236100000, 344200000

Offset: 1

Author

Jonathan Ayres (jonathan.ayres(AT)btinternet.com), Nov 17 2001

Keywords

• base
• nonn

Comments

Are there any terms > 3442 that are not just a previous term followed by zeros?

Concerning this question, see the a-file with terms up to 10^6 expressed in the corresponding base for similar sequences in base 2 to 37. - Michel Marcus, Dec 17 2015

Examples

			n = 655 is in the sequence because f(655) = 6*55 + 65*5 = 330 + 325 = 655.
		

Links

• Sean A. Irvine, Table of n, a(n) for n = 1..47
• Michel Marcus, Similar sequences in base b=2 to 37

Programs

Mathematica

f[n_] := Block[{a = {}, e = IntegerLength@ n - 1, k}, Do[AppendTo[a, #*(n - #*10^(e - k)) &@ Floor[n/10^(e - k)]], {k, 0, e - 1}]; Total@ a]; Select[0, Range[10^6], f@ # == # &] (* Michael De Vlieger, Dec 18 2015 *)

PARI

isok(n) = n == sum(k=1, #Str(n), (n\10^k)*(n % 10^k)); \\ Michel Marcus, Dec 16 2015

Formula

Conjectures from Colin Barker, Jun 18 2019: (Start)

G.f.: x*(655 + 1461*x + 1642*x^2 + 2361*x^3 + 3442*x^4) / (1 - 10*x^5).

a(n) = 10*a(n-5) for n>5.

(End)

Extensions

0 added by Rémy Sigrist, May 21 2021

More terms from Sean A. Irvine, Sep 12 2023