A258682 -id:A258682 - cp's OEIS frontend

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

Patrick De Geest

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

			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)

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.

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

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

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

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

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

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

Python
```
# see linked program
```

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

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

A189056 Numbers having in decimal representation at least one common digit with their squares.

Original entry on oeis.org

0, 1, 5, 6, 10, 11, 12, 13, 14, 15, 16, 19, 20, 21, 23, 25, 26, 27, 28, 30, 31, 32, 35, 36, 37, 40, 41, 42, 43, 45, 46, 48, 49, 50, 51, 52, 55, 56, 60, 61, 63, 64, 65, 66, 68, 69, 70, 71, 73, 74, 75, 76, 78, 80, 81, 82, 83, 85, 86, 87, 89, 90, 91, 93, 95, 96, 97, 98, 99, 100, 101, 102, 103, 104, 105, 106, 107, 108, 109, 110, 111, 112, 113, 114, 115, 116, 117, 118, 119, 120

Offset: 1

Reinhard Zumkeller, Apr 16 2011

Complement of A029783; A076493(a(n)) > 0.

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

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A029783, A076493, A258682.

a189056 n = a189056_list !! (n-1)
a189056_list = 0 : filter (\x -> a258682 x /= x ^ 2) [1..]
-- Reinhard Zumkeller, Jun 07 2015, Apr 16 2011

Select[Range[0,120],Length[Intersection[IntegerDigits[#], IntegerDigits[ #^2]]]>0&] (* Harvey P. Dale, Aug 28 2012 *)

isok(n) = (n==0) || #setintersect(Set(digits(n)), Set(digits(n^2))); \\ Michel Marcus, Jun 13 2015

A387070 Remove every digit that appears in n from the decimal representation of n^2. If no digits remain, set a(n) = 0.

Original entry on oeis.org

0, 0, 4, 9, 16, 2, 3, 49, 64, 81, 0, 2, 44, 69, 96, 22, 25, 289, 324, 36, 4, 44, 484, 59, 576, 6, 7, 9, 74, 841, 9, 96, 104, 1089, 1156, 122, 129, 169, 1444, 1521, 16, 68, 176, 189, 1936, 202, 211, 2209, 230, 201, 2, 260, 704, 2809, 2916, 302, 313, 3249, 3364, 3481, 3, 372, 3844, 99, 9, 422, 435

Offset: 0

Ali Sada, Aug 15 2025

			a(25) = 6 since 25^2 = 625 and once we remove the 2 and 5, we are left with 6.
a(26) = 7 since 26^2 = 676 and once we remove the 6, we are left with 7.

Cf. A029793, A258682.

a[n_] := FromDigits[Select[IntegerDigits[n^2], FreeQ[IntegerDigits[n], #] &]]; Array[a, 100, 0] (* Amiram Eldar, Aug 16 2025 *)

a(n)={my(S=Set(digits(n))); fromdigits(select(x->!setsearch(S,x), digits(n^2)))} \\ Andrew Howroyd, Aug 15 2025

def A387070(k):
    s = set(str(k))
    t = "".join(d for d in str(k**2) if d not in s)
    return int(t) if t != "" else 0
print([A387070(n) for n in range(67)]) # Michael S. Branicky, Aug 16 2025

a(n) = A258682(n) for n <= 19.
From David A. Corneth, Aug 16 2025: (Start)
a(A029793(k)) = 0.
a(A029783(k)) = A029783(k)^2. (End)

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

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A189056 Numbers having in decimal representation at least one common digit with their squares.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Haskell

Mathematica

PARI

A387070 Remove every digit that appears in n from the decimal representation of n^2. If no digits remain, set a(n) = 0.

Views

Author

Keywords

Examples

Crossrefs

Programs

Mathematica

PARI

Python

Formula