A245584 Let f(m) put the leftmost digit of the positive integer m at its end; a(n) is the sequence of all positive integers m with f^2(m)=f(m^2).

1, 2, 3, 12, 122, 1222, 12222, 122222, 1222222, 12222222, 122222222

Offset: 1

Author

Reiner Moewald, Jul 26 2014

Keywords

• nonn
• base
• more

Examples

			122^2=14884 and 221^2=48841.
		

Crossrefs

Cf. A045878, A090843.

Programs

Mathematica

f[m_Integer] := Module[{w}, w := IntegerDigits[m]; FromDigits[Rest[AppendTo[w, First[w]]]]]; a245584[n_Integer] :=
Select[Range[n], If[f[#]^2 == f[#^2] && ! Mod[#, 10] == 0, True, False] &]; a245584[10^5] (* Michael De Vlieger, Aug 17 2014 *)

Python

import math
max = 10000
print('los')
for n in range(1, max):
   nst = str(n*n)
   nnewst = nst[1:] + nst[0]
   d = int(nnewst)
   e = int(math.sqrt(d))
   est = str(e)
   enewst = est[len(est)-1] + est[:len(est)-1]
   if (e * e == d) and (nnewst[0] != "0") and (str(n) == enewst):
      print(n, '  ',  e)
print('End.')

Formula

One can easily prove that all integers of the form 12...2 are elements of the sequence.