A054038 Numbers k such that k^2 contains every digit at least once.

32043, 32286, 33144, 35172, 35337, 35757, 35853, 37176, 37905, 38772, 39147, 39336, 40545, 42744, 43902, 44016, 45567, 45624, 46587, 48852, 49314, 49353, 50706, 53976, 54918, 55446, 55524, 55581, 55626, 56532, 57321, 58413, 58455

Offset: 1

Author

Asher Auel, Feb 28 2000

Keywords

• nonn
• base

Comments

There are 87 terms < 10^5; these are the n such that n^2 uses each digit exactly once. - David Wasserman, Feb 03 2005

The squares in this sequence are in A190682. - Bruno Berselli, May 23 2011

References

• J.-M. De Koninck and A. Mercier, 1001 Problemes en Theorie Classique Des Nombres, Problem 239 pp. 39; 178, Ellipses Paris 2004.

Links

• Giovanni Resta, Table of n, a(n) for n = 1..10000 (first 4866 terms from Klaus Brockhaus)

Crossrefs

Cf. A016069, A054031, A054032, A054033, A054034, A054035, A054036, A054037, A054039.

Programs

Magma

IsA054038:=func< n | Seqset(Intseq(n^2)) eq {0,1,2,3,4,5,6,7,8,9} >; [ n: n in [1..60000] | IsA054038(n) ]; // Klaus Brockhaus, May 16 2011

Maple

f := []; for i from 0 to 200 do if nops({op(convert(i^2,base,10))})=10 then f := [op(f),i] fi; od; f;

Mathematica

A050278 = Select[FromDigits@#&/@Permutations[Range[0, 9], {10}], # > 10^9 &]; Sqrt[Select[A050278, IntegerQ[Sqrt[#]] &]] (* Alonso del Arte, Jun 18 2011, based on a program by Robert G. Wilson v *)
Select[Sqrt[#]&/@FromDigits/@Select[Permutations[Range[0,9]],#[[1]]>0&], IntegerQ] (* Harvey P. Dale, May 26 2016 *)

PARI

is(n)=#vecsort(Vec(Str(n^2)),,8)==10 \\ Charles R Greathouse IV, Jun 18 2011

Python

def ok(n): return len(set(str(n**2))) == 10
print([k for k in range(10**5) if ok(k)]) # Michael S. Branicky, Dec 23 2022

Extensions

More terms from David Wasserman, Feb 03 2005