A112736 Numbers whose square is exclusionary.

2, 3, 4, 7, 8, 9, 17, 18, 24, 29, 34, 38, 39, 47, 53, 54, 57, 58, 59, 62, 67, 72, 79, 84, 92, 94, 157, 158, 173, 187, 192, 194, 209, 237, 238, 247, 253, 257, 259, 307, 314, 349, 359, 409, 437, 459, 467, 547, 567, 612, 638, 659, 672, 673, 689, 712, 729, 738, 739, 749

Offset: 1

Author

Lekraj Beedassy, Sep 16 2005

Keywords

• nonn
• base
• fini
• full

Comments

The number m with no repeated digits has an exclusionary square m^2 if the latter is made up of digits not appearing in m. For the corresponding exclusionary squares see A112735.

a(49) = 567 and a(68) = 854 are the only two numbers k such that the equation k^2 = m uses only once each of the digits 1 to 9 (reference David Wells). Exactly: 567^2 = 321489, and, 854^2 = 729316. - Bernard Schott, Dec 20 2021

Examples

			409^2 = 167281 and the square 167281 is made up of digits not appearing in 409, hence 409 is a term.
		

References

• H. Ibstedt, Solution to Problem 2623, "Exclusionary Powers", pp. 346-9, Journal of Recreational Mathematics, Vol. 32 No.4 2003-4 Baywood NY.
• David Wells, The Penguin Dictionary of Curious and Interesting Numbers, Revised Edition, 1997, page 144, entry 567.

Links

• Giovanni Resta, Table of n, a(n) for n = 1..142 (full sequence)

Crossrefs

Cf. A000290, A112321, A112735.

This is a subsequence of A029783 (Digits of n are not present in n^2) of numbers with all different digits. The sequence A059930 (Numbers n such that n and n^2 combined use different digits) is a subsequence of this sequence.

Programs

Mathematica

Select[Range[1000], Intersection[IntegerDigits[ # ], IntegerDigits[ #^2]] == {} && Length[Union[IntegerDigits[ # ]]] == Length[IntegerDigits[ # ]] &] (* Tanya Khovanova, Dec 25 2006 *)

Extensions

More terms from Tanya Khovanova, Dec 25 2006