A282767 n/3 analog of Keith numbers.

45, 609, 1218, 1827, 3213, 21309, 28206, 29319, 31917, 39333, 47337, 78666, 102090, 117999, 204180, 406437, 302867592, 4507146801, 5440407522

Offset: 1

Author

Paolo P. Lava, Feb 27 2017

Keywords

• nonn
• base
• more

Comments

Like Keith numbers but starting from n/3 digits to reach n.

Consider the digits of n/3. Take their sum and repeat the process deleting the first addend and adding the previous sum. The sequence lists the numbers that after some iterations reach a sum equal to themselves.

If it exists, a(20) > 10^12. - Lars Blomberg Mar 13 2017

Examples

			609/3 = 203:
2 + 0 + 3 = 5;
0 + 3 + 5 = 8;
3 + 5 + 8 = 16;
5 + 8 + 16 = 29;
8 + 16 + 29 = 53;
16 + 29 + 53 = 98;
29 + 53 + 98 = 180;
53 + 98 + 180 = 331;
98 + 180 + 331 = 609.
		

Crossrefs

Cf. A282757 - A282765, A282766, A282768, A282769.

Programs

Maple

with(numtheory): P:=proc(q,h,w) local a, b, k, n, t, v; v:=array(1..h);
for n from 1/w by 1/w to q do a:=w*n; b:=ilog10(a)+1; if b>1 then
for k from 1 to b do v[b-k+1]:=(a mod 10); a:=trunc(a/10); od; t:=b+1; v[t]:=add(v[k], k=1..b);
while v[t]
				

Mathematica

With[{n = 3}, Select[Range[10 n, 10^6, n], Function[k, Last@ NestWhile[Append[Rest@ #, Total@ #] &, IntegerDigits[k/n], Total@ # <= k &] == k]]] (* Michael De Vlieger, Feb 27 2017 *)

Extensions

a(17)-a(19) from Lars Blomberg, Mar 13 2017