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

1,1

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

Links

Table of n, a(n) for n=1..19.

Example

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.

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]<n do t:=t+1; v[t]:=add(v[k], k=t-b..t-1); od;
if v[t]=n then print(n); fi; fi; od; end: P(10^6, 1000, 1/3);

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 *)

Crossrefs

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

Sequence in context: A160838 A189350 A090024 * A372978 A160234 A110691

Adjacent sequences: A282764 A282765 A282766 * A282768 A282769 A282770

Keyword

nonn,base,more

Author

Paolo P. Lava, Feb 27 2017

Extensions

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

Status

approved