A045540 Numbers whose square contains an equal number of each digit that it contains.

1, 2, 3, 4, 5, 6, 7, 8, 9, 13, 14, 16, 17, 18, 19, 23, 24, 25, 27, 28, 29, 31, 32, 33, 36, 37, 42, 43, 44, 48, 49, 51, 52, 53, 54, 55, 57, 59, 61, 64, 66, 69, 71, 72, 73, 74, 78, 79, 82, 84, 86, 87, 88, 89, 93, 95, 96, 98, 99, 113, 116, 117, 118, 124, 126, 128, 133, 134, 136

Offset: 1

Author

Erich Friedman

Keywords

• base
• nonn

Comments

The sequence is expected to be infinite. Heuristically, if m is divisible by 10 there should be approximately constant * 10^(m/2)/m^(9/2) m-digit squares where all 10 digits have frequency m/10. - Robert Israel, Aug 14 2015

Links

• Robert Israel, Table of n, a(n) for n = 1..3000
• P. De Geest, Numbers whose digits occur with same frequency

Crossrefs

Cf. A052046, A052047, A052048, A052049, A052050, A052051, A052052, A052060.

Programs

Maple

filter:= proc(n) local x,i,P;
P:= add(x^i, i=convert(n^2,base,10));
nops({coeffs(P,x)}) = 1
end proc:
select(filter, [$1..10^4]); # Robert Israel, Aug 14 2015

Mathematica

t={}; Do[If[Length[DeleteDuplicates[Transpose[Tally[IntegerDigits[n^2]]][[2]]]]==1,AppendTo[t,n]],{n,136}]; t (* Jayanta Basu, May 10 2013 *)