A384296 Square numbers whose iterative sums of digits are squares.

0, 1, 4, 9, 36, 81, 100, 121, 144, 225, 324, 400, 441, 900, 1521, 2025, 2304, 2601, 3600, 8100, 10000, 10201, 10404, 11025, 12100, 12321, 14400, 22500, 32400, 40000, 40401, 44100, 62001, 69696, 90000, 101124, 103041, 121104, 123201, 149769, 152100, 173889, 178929, 199809, 202500, 230400, 251001

Offset: 1

Author

Huaineng He, May 24 2025

Keywords

• nonn
• base

Examples

			10201 = 101^2. 1+0+2+0+1 = 4, 4 = 2^2.
178929 = 423^2. 1+7+8+9+2+9 = 36, 36 = 6^2. 3+6 = 9, 9 = 3^2.
		

Crossrefs

Cf. A004159, A117676, A070027.

A subsequence of A053057.

Programs

Maple

filter:= proc(n) local L,i,t;
  t:= n;
  do
    if not issqr(t) then return false fi;
    if t < 10 then return true fi;
    t:= convert(convert(t,base,10),`+`)
  od
end proc:
select(filter, [seq(i^2,i=0..1000)]); # Robert Israel, May 25 2025

Mathematica

q[n_] := AllTrue[FixedPointList[DigitSum, n], IntegerQ[Sqrt[#]] &]; Select[Range[0, 500]^2, q] (* Amiram Eldar, May 25 2025 *)