A059930 Numbers n such that n and n^2 combined use different digits.

2, 3, 4, 7, 8, 9, 17, 18, 24, 29, 53, 54, 57, 59, 72, 79, 84, 209, 259, 567, 807, 854

Offset

1,1

Comments

There are exactly 22 solutions in base 10.

More precisely: the concatenation of n and n^2 does not contain any digit twice. - M. F. Hasler, Oct 16 2018

a(20) = 567 and a(22) = 854 are the only two numbers k such that k and k^2 combined use each of the digits 1 to 9 exactly once (reference David Wells): 567^2 = 321489 and 854^2 = 729316. - Bernard Schott, Mar 23 2021

References

M. Kraitchik, Mathematical Recreations, p. 48, Problem 12. - From N. J. A. Sloane, Mar 15 2013

David Wells, The Penguin Dictionary of Curious and Interesting Numbers, Revised Edition, 1997, page 144, entry 567.

Links

Table of n, a(n) for n=1..22.

Maple

M:=1000;
a1:=[]; a2:=[];
for n from 1 to M do
# are digits of n and n^2 distinct?
t1:=convert(n, base, 10);
t2:=convert(n^2, base, 10);
s3:={op(t1), op(t2)};
if nops(t1)+nops(t2) = nops(s3) then a1:=[op(a1), n]; a2:=[op(a2), n^2]; fi;
od:
a1; a2;
# N. J. A. Sloane, Mar 15 2013

Mathematica

Select[Range[10000], Intersection[IntegerDigits[ # ], IntegerDigits[ #^2]] == {} && Length[Union[IntegerDigits[ # ], IntegerDigits[ #^2]]] == Length[IntegerDigits[ # ]] + Length[IntegerDigits[ #^2]] &] (* Tanya Khovanova, Dec 25 2006 *)
Select[Range[10^3], Union@ Tally[Flatten@ IntegerDigits@ {#, #^2}][[All, -1]] == {1} &] (* Michael De Vlieger, Oct 17 2018 *)

Prog

(PARI) select( is(n)=#Set(Vecsmall(n=Str(n, n^2)))==#n, [1..999]) \\ M. F. Hasler, Oct 16 2018

Crossrefs

Cf. A059931, A029783 (digits of n are not present in n^2), A112736 (numbers whose squares are exclusionary).

Sequence in context: A340324 A029783 A112736 * A125965 A111116 A113318

Adjacent sequences: A059927 A059928 A059929 * A059931 A059932 A059933

Keyword

nonn,base,fini,full

Author

Patrick De Geest, Feb 15 2001

Status

approved