A189056 -id:A189056 - 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

A076493 Number of common (distinct) decimal digits of n and n^2.

Original entry on oeis.org

1, 1, 0, 0, 0, 1, 1, 0, 0, 0, 2, 1, 1, 1, 1, 1, 1, 0, 0, 1, 1, 1, 0, 1, 0, 2, 1, 2, 1, 0, 1, 1, 1, 0, 0, 1, 1, 1, 0, 0, 1, 1, 1, 1, 0, 1, 1, 0, 1, 1, 2, 1, 1, 0, 0, 1, 1, 0, 0, 0, 2, 1, 0, 2, 2, 1, 1, 0, 1, 1, 1, 1, 0, 1, 2, 1, 2, 0, 1, 0, 1, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 0, 2, 2, 1, 1, 1, 2, 2, 2, 2, 2, 3, 2, 1, 1, 1

Offset: 0

Labos Elemer, Oct 21 2002

a(A029783(n)) = 0, a(A189056(n)) > 0; 0 <= a(n) <= 10, see example for first occurrences. [Reinhard Zumkeller, Apr 16 2011]

			a(2) = #{} = 0: 2^2 = 4;
a(0) = #{0} = 1: 0^2 = 0;
a(10) = #{0,1} = 2: 10^2 = 100;
a(105) = #{0,1,5} = 3: 105^2 = 11025;
a(1025) = #{0,1,2,5} = 4: 1025^2 = 1050625;
a(10245) = #{0,1,2,4,5} = 5: 10245^2 = 104960025;
a(102384) = #{0,1,2,3,4,8} = 6: 102384^2 = 10482483456;
a(1023456789) = #{0 .. 9} = 10: 1023456789^2 = 1047463798950190521.

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000

Cf. A006880, A076489-A076491.

Cf. A000290, A043537.

import Data.List (intersect, nub)
import Data.Function (on)
a076493 n = length $ (intersect `on` nub . show) n (n^2)
-- Reinhard Zumkeller, Apr 16 2011

Table[Length[Intersection[IntegerDigits[n], IntegerDigits[n^2]]], {n, 1, 100}]

Initial 1 added and offset adjusted by Reinhard Zumkeller, Apr 16 2011

A253600 Smallest exponent k>1 such that n and n^k have some digits in common.

Original entry on oeis.org

2, 2, 5, 5, 3, 2, 2, 5, 5, 3, 2, 2, 2, 2, 2, 2, 2, 3, 3, 2, 2, 2, 4, 2, 3, 2, 2, 2, 2, 3, 2, 2, 2, 3, 3, 2, 2, 2, 3, 3, 2, 2, 2, 2, 3, 2, 2, 4, 2, 2, 2, 2, 2, 5, 3, 2, 2, 3, 3, 3, 2, 2, 3, 2, 2, 2, 2, 3, 2, 2, 2, 2, 3, 2, 2, 2, 2, 5, 2, 3, 2, 2, 2, 2, 3, 2, 2

Offset: 0

Michel Marcus, Jan 05 2015

For all n, n^5-n is divisible by 10, and so n^5 == n (mod 10). Thus a(n) <= 5 for all n. - Tom Edgar, Jan 06 2015

			For n=2, 2^k has no digit in common with 2 until k reaches 5 to give 32, hence a(2)=5.

Robert Israel, Table of n, a(n) for n = 0..10000

Cf. sequences where a(n)=k: A103173 (k=5), A189056 (k=2), A253601 (k=3), A253602 (k=4).

Cf. A373203.

f:= proc(n) local L,k;
 L:= convert(convert(n,base,10),set);
 for k from 2 do
   if convert(convert(n^k,base,10),set) intersect L <> {} then
     return k
   fi
 od
end proc:
map(f, [$0..100]); # Robert Israel, Mar 17 2020

seq={};Do[k=1;Until[ContainsAny[IntegerDigits[n],IntegerDigits[n^k]],k++];AppendTo[seq,k] ,{n,0,86}];seq (* James C. McMahon, Jun 04 2024 *)

a(n) = {sd = Set(vecsort(digits(n))); k=2; while (#setintersect(sd, Set(vecsort(digits(n^k)))) == 0, k++); k;}

A258682 Remove from n^2 all digits of n from left to right, in decimal representation.

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, 40, 44, 484, 59, 576, 6, 76, 9, 74, 841, 90, 96, 104, 1089, 1156, 122, 129, 169, 1444, 1521, 160, 681, 176, 189, 1936, 202, 211, 2209, 230, 201, 20, 260, 704, 2809, 2916, 302, 313

Offset: 0

Reinhard Zumkeller, Jun 07 2015

a(A029783(n)) = A029783(n)^2; a(A189056(n)) < A189056(n)^2;

a(A178501(n)) = 0.

			.    n   n^2 |  a(n)
. -----------+------
.   20   400 |    40
.   21   441 |    44
.   22   484 |   484
.   23   529 |    59
.   24   576 |   576
.   25   625 |     6
.   26   676 |    76  not 67, as digits of n are to be removed
.   27   729 |     9                                  from left to right
.   28   784 |    74
.   29   841 |   841
.   30   900 |    90  .

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000

Cf. A000290, A029783, A189056, A178501.

import Data.List (delete)
a258682 n = read ('0' : minus (show (n ^ 2)) (show n)) :: Int  where
   minus [] _  = []
   minus us [] = us
   minus (u:us) vs | elem u vs = minus us $ delete u vs
                   | otherwise = u : minus us vs

a[n_]:=Module[{idn=IntegerDigits[n],idnn=IntegerDigits[n^2],idn1},
idn1=DeleteCases[idn,m_/;MemberQ[Complement[idn,idnn],m]];
Do[If[First[Position[idnn,idn1[[i]]]]!= {},idnn=Delete[idnn,First[Position[idnn,idn1[[i]]]]]],
{i,1,Length[idn1]}];FromDigits[idnn]]//Quiet;
a/@Range[0,56] (* Ivan N. Ianakiev, Jun 11 2015 *)

A253601 Numbers such that the smallest exponent for n and n^k to have common digits is 3.

Original entry on oeis.org

4, 9, 17, 18, 24, 29, 33, 34, 38, 39, 44, 54, 57, 58, 59, 62, 67, 72, 79, 84, 88, 94, 144, 158, 173, 194, 209, 212, 224, 237, 238, 244, 247, 253, 257, 259, 307, 313, 314, 333, 334, 338, 349, 353, 359, 377, 388, 404, 409, 424, 437, 444, 447, 449, 454, 459, 467

Offset: 1

Michel Marcus, Jan 05 2015

			4^2=16 has no digits in common with 4, but 4^3=64 has some, so 4 is in the sequence.

Iain Fox, Table of n, a(n) for n = 1..10000

Cf. A103173, A189056, A253600, A253602.

a253600(n) = {sd = Set(vecsort(digits(n))); k=2; while (#setintersect(sd, Set(vecsort(digits(n^k)))) == 0, k++); k;}
isok(n) = a253600(n) == 3;

is(n) = my(d(k)=Set(digits(n^k))); !#setintersect(d(1), d(2)) && #setintersect(d(1), d(3)) \\ Iain Fox, Aug 07 2018

A253602 Numbers n such that the smallest exponent k for n and n^k to have common digits is 4.

Original entry on oeis.org

22, 47, 92, 157, 187, 188, 192, 552, 558, 577, 707, 772, 922, 2522, 8338, 17177, 66888, 575757, 929522, 1717177, 8888588

Offset: 1

Michel Marcus, Jan 05 2015

			22^2=484 and 22^3=10648 have no digits in common with 22, but 22^4=234256 has some, so 22 is in the sequence.

Cf. A029790, A103173, A189056, A253600, A253601.

a253600(n) = {sd = Set(vecsort(digits(n))); k=2; while (#setintersect(sd, Set(vecsort(digits(n^k)))) == 0, k++); k;}
isok(n) = a253600(n) == 4;

A326418 Nonnegative numbers k such that, in decimal representation, the subsequence of digits of k^2 occupying an odd position is equal to the digits of k.

Original entry on oeis.org

0, 1, 5, 6, 10, 11, 50, 60, 76, 100, 105, 110, 500, 501, 505, 506, 600, 605, 756, 760, 826, 1000, 1001, 1050, 1100, 5000, 5010, 5050, 5060, 5941, 6000, 6050, 7560, 7600, 8260, 10000, 10005, 10010, 10500, 10505, 11000, 12731

Offset: 1

Riccardo Pietro Giovambattista Mazzei and Matteo Albanese, Sep 28 2019

If k is in the sequence then so is 10*k. - David A. Corneth, Sep 29 2019

No term starts with the digit 2. - Chai Wah Wu, Apr 04 2023

			5^2 = 25, whose first digit is 5, hence 5 is a term of the sequence.
11^2 = 121, whose first and third digit are (1, 1), hence 11 is a term of the sequence.
756^2 = 571536, whose digits in odd positions - starting from the least significant one - are (7, 5, 6), hence 756 is a term of the sequence.

David A. Corneth, Table of n, a(n) for n = 1..2361 (terms < 10^15; first 485 terms from Charles R Greathouse IV)

Subsequence of A008851, A029772, A046829, A064827, A052212, A189056.

Select[Range[0, 13000], Reverse@ #[[-Range[1, Length@ #, 2]]] &@ IntegerDigits[#^2] === IntegerDigits[#] &] (* Michael De Vlieger, Oct 06 2019 *)

isok(n) = my(d=Vecrev(digits(n^2))); fromdigits(Vecrev(vector((#d+1)\2, k, d[2*k-1]))) == n; \\ Michel Marcus, Oct 01 2019

def ok(n): s = str(n*n); return n == int("".join(s[1-len(s)%2::2]))
print(list(filter(ok, range(13000)))) # Michael S. Branicky, Sep 10 2021

cp's OEIS Frontend

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

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A076493 Number of common (distinct) decimal digits of n and n^2.

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A253600 Smallest exponent k>1 such that n and n^k have some digits in common.

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Maple

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A258682 Remove from n^2 all digits of n from left to right, in decimal representation.

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Haskell

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A253601 Numbers such that the smallest exponent for n and n^k to have common digits is 3.

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PARI

PARI

A253602 Numbers n such that the smallest exponent k for n and n^k to have common digits is 4.

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PARI

A326418 Nonnegative numbers k such that, in decimal representation, the subsequence of digits of k^2 occupying an odd position is equal to the digits of k.

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Programs

Mathematica