A029783 Exclusionary squares: numbers n such that no digit of n is present in n^2.

2, 3, 4, 7, 8, 9, 17, 18, 22, 24, 29, 33, 34, 38, 39, 44, 47, 53, 54, 57, 58, 59, 62, 67, 72, 77, 79, 84, 88, 92, 94, 144, 157, 158, 173, 187, 188, 192, 194, 209, 212, 224, 237, 238, 244, 247, 253, 257, 259, 307, 313, 314, 333, 334, 338, 349, 353, 359

Offset: 1

Author

Patrick De Geest

Keywords

• nonn
• base

Comments

Complement of A189056; A076493(a(n)) = 0. - Reinhard Zumkeller, Apr 16 2011

A258682(a(n)) = a(n)^2. - Reinhard Zumkeller, Jun 07 2015

a(78) = 567 and a(112) = 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 (see A059930). - Bernard Schott, Jan 28 2021

Examples

			From _M. F. Hasler_, Oct 16 2018: (Start)
It is easy to construct infinite subsequences of the form S(a,b)(n) = a*R(n) + b, where R(n) = (10^n-1)/9 is the repunit of length n. Among these are:
S(3,0) = (3, 33, 333, ...), S(3,1) = (4, 34, 334, 3334, ...), S(3,5) = (8, 38, 338, ...), also b = 26, 44, 434, ... (with a = 3); S(6,1) = (7, 67, 667, ...), S(6,6) = (72, 672, 6672, ...) (excluding n=1), S(6,7) = (673, 6673, ...) (excluding also n=2 here), S(6,-7) = (59, 659, 6659, ...), and others. (End)
		

References

• Clifford A. Pickover, A Passion for Mathematics, Wiley, 2005; see p. 60.
• 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..10000 (first 1000 terms from Reinhard Zumkeller)
• Michael S. Branicky, Python program
• Cliff Pickover et al, Exclusionary Squares and Cubes, rec.puzzles topic on google groups, January 2002

Crossrefs

Cf. A059930 (n and n^2 use different digits), A112736 (numbers whose squares are exclusionary).

Cf. A029784, A029785, A029786, A111116, A113316, A189056, A076493, A258682.

Programs

Haskell

a029783 n = a029783_list !! (n-1)
a029783_list = filter (\x -> a258682 x == x ^ 2) [1..]
-- Reinhard Zumkeller, Jun 07 2015, Apr 16 2011

Mathematica

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

PARI

is_A029783(n)=!#setintersect(Set(digits(n)),Set(digits(n^2))) \\ M. F. Hasler, Oct 16 2018

Python

# see linked program

Python

from itertools import count, islice
def A029783_gen(startvalue=0): # generator of terms >= startvalue
    return filter(lambda n:not set(str(n))&set(str(n**2)),count(max(startvalue,0)))
A029783_list = list(islice(A029783_gen(),30)) # Chai Wah Wu, Feb 12 2023

Extensions

Definition slightly reworded at the suggestion of Franklin T. Adams-Watters by M. F. Hasler, Oct 16 2018