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

1,1

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

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

Example

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. These are:
S(3,0) = (3, 33, 333, ...), S(3,1) = (4, 34, 334, 3334, ...), S(3,5) = (8, 38, 338, ...), S(6,0) = (6, 66, 666, ...), 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) and S(6,-7) = (59, 659, 6659, ...). (End)

Mathematica

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

Prog

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

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.

Sequence in context: A058075 A243495 A340324 * A112736 A059930 A125965

Adjacent sequences: A029780 A029781 A029782 * A029784 A029785 A029786

Keyword

nonn,base

Author

Patrick De Geest

Extensions

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

Status

approved