A274769 Square analog to Keith numbers.

1, 9, 37, 40, 43, 62, 70, 74, 160, 1264, 1952, 2847, 12799, 16368, 16584, 42696, 83793, 97415, 182011, 352401, 889871, 925356, 1868971, 1881643, 3661621, 7645852, 15033350, 21655382, 63288912, 88192007, 158924174, 381693521, 792090500, 2025078249, 2539401141

Offset: 1

Author

Paolo P. Lava, Jul 06 2016

Keywords

• nonn
• base

Comments

Like Keith numbers but starting from n^2 digits to reach n.

Consider the digits of the square of a number n. 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.

Examples

			1264^2 = 1597696 :
1 + 5 + 9 + 7 + 6 + 9 + 6 = 43;
5 + 9 + 7 + 6 + 9 + 6 + 43 = 85;
9 + 7 + 6 + 9 + 6 + 43 + 85 = 165;
7 + 6 + 9 + 6 + 43 + 85 + 165 = 321;
6 + 9 + 6 + 43 + 85 + 165 + 321 = 635;
9 + 6 + 43 + 85 + 165 + 321 + 635 = 1264.
		

Crossrefs

Cf. A007629, A246544, A263534, A274770.

Programs

Maple

with(numtheory): P:=proc(q, h) local a,b,k,n,t,v; v:=array(1..h);
for n from 1 to q do b:=n^2; a:=[];
for k from 1 to ilog10(b)+1 do a:=[(b mod 10),op(a)]; b:=trunc(b/10); od;
for k from 1 to nops(a) do v[k]:=a[k]; od; b:=ilog10(n^2)+1;
t:=nops(a)+1; v[t]:=add(v[k], k=1..b); while v[t]
				

Mathematica

Select[Range[10^6], Function[n, Module[{d = IntegerDigits[n^2], s, k = 0}, s = Total@ d; While[s < n, AppendTo[d, s]; k++; s = 2 s - d[[k]]]; s == n]]] (* Michael De Vlieger, Feb 22 2017, after T. D. Noe at A007629 *)
(* function keithQ[ ] is defined in A007629 *)
a274769[n_] := Join[{1, 9}, Select[Range[10, n], keithQ[#, 2]&]]
a274769[10^6] (* Hartmut F. W. Hoft, Jun 02 2021 *)

Extensions

a(32)-a(35) from Giovanni Resta, Jul 08 2016