A112736 -id:A112736 - 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

A059930 Numbers n such that n and n^2 combined use different digits.

Original entry on oeis.org

2, 3, 4, 7, 8, 9, 17, 18, 24, 29, 53, 54, 57, 59, 72, 79, 84, 209, 259, 567, 807, 854

Offset: 1

Patrick De Geest, Feb 15 2001

There are exactly 22 solutions in base 10.
More precisely: the concatenation of n and n^2 does not contain any digit twice. - M. F. Hasler, Oct 16 2018
a(20) = 567 and a(22) = 854 are the only two numbers k such that k and k^2 combined use each of the digits 1 to 9 exactly once (reference David Wells): 567^2 = 321489 and 854^2 = 729316. - Bernard Schott, Mar 23 2021

M. Kraitchik, Mathematical Recreations, p. 48, Problem 12. - From N. J. A. Sloane, Mar 15 2013
David Wells, The Penguin Dictionary of Curious and Interesting Numbers, Revised Edition, 1997, page 144, entry 567.

Cf. A059931, A029783 (digits of n are not present in n^2), A112736 (numbers whose squares are exclusionary).

M:=1000;
a1:=[]; a2:=[];
for n from 1 to M do
# are digits of n and n^2 distinct?
t1:=convert(n,base,10);
t2:=convert(n^2,base,10);
s3:={op(t1),op(t2)};
if nops(t1)+nops(t2) = nops(s3) then a1:=[op(a1),n]; a2:=[op(a2),n^2]; fi;
od:
a1; a2;
# N. J. A. Sloane, Mar 15 2013

Select[Range[10000], Intersection[IntegerDigits[ # ], IntegerDigits[ #^2]] == {} && Length[Union[IntegerDigits[ # ], IntegerDigits[ #^2]]] == Length[IntegerDigits[ # ]] + Length[IntegerDigits[ #^2]] &] (* Tanya Khovanova, Dec 25 2006 *)
Select[Range[10^3], Union@ Tally[Flatten@ IntegerDigits@ {#, #^2}][[All, -1]] == {1} &] (* Michael De Vlieger, Oct 17 2018 *)

select( is(n)=#Set(Vecsmall(n=Str(n,n^2)))==#n, [1..999]) \\ M. F. Hasler, Oct 16 2018

A113951 Largest number whose n-th power is exclusionary (or 0 if no such number exists).

Original entry on oeis.org

639172, 7658, 2673, 0, 92, 93, 712, 0, 18, 12, 4, 0, 37, 0, 9, 0, 0, 3, 4, 0, 7, 2, 7, 0, 8, 3, 9, 0, 0, 0, 0, 0, 3, 2, 2, 0, 0, 7, 3, 0, 2, 0, 0, 0, 2, 0, 0, 0, 3, 0, 0, 0, 2, 3, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 2, 3

Offset: 2

Lekraj Beedassy, Nov 09 2005

The number m with no repeated digits has an exclusionary n-th power m^n if the latter is made up of digits not appearing in m. For the corresponding m^n see A113952. In principle, no exclusionary n-th power exists for n == 1 (mod 4) = A016813.

After a(84) = 3, the next nonzero term is a(168) = 2, where 168 is the last known term in A034293. - Michael S. Branicky, Aug 28 2021

			a(4) = 2673 because no number with distinct digits beyond 2673 has a 4th power that shares no digit in common with it (see A111116). Here we have 2673^4 = 51050010415041.

H. Ibstedt, Solution to Problem 2623, "Exclusionary Powers", pp. 346-9 Journal of Recreational Mathematics, Vol. 32 No.4 2003-4, Baywood NY.

Michael S. Branicky, Table of n, a(n) for n = 2..225

Cf. A109135, A112736, A112994, A113318, A034293.

from itertools import combinations, permutations
def no_repeated_digits():
    for d in range(1, 11):
        for p in permutations("0123456789", d):
            if p[0] == '0': continue
            yield int("".join(p))
def a(n):
    m = 0
    for k in no_repeated_digits():
        if set(str(k)) & set(str(k**n)) == set():
            m = max(m, k)
    return m
for n in range(2, 4): print(a(n), end=", ") # Michael S. Branicky, Aug 28 2021

a(34), a(39), a(40) corrected by and a(43) and beyond from Michael S. Branicky, Aug 28 2021

A112735 Exclusionary squares.

Original entry on oeis.org

4, 9, 16, 49, 64, 81, 289, 324, 576, 841, 1156, 1444, 1521, 2209, 2809, 2916, 3249, 3364, 3481, 3844, 4489, 5184, 6241, 7056, 8464, 8836, 24649, 24964, 29929, 34969, 36864, 37636, 43681, 56169, 56644, 61009, 64009, 66049, 67081, 94249, 98596

Offset: 1

Lekraj Beedassy, Sep 16 2005

An exclusionary square m^2 is one sharing no digit in common with its root m made up of distinct digits. The associated root is given by A112736.

The largest term is 639172^2 = 408540845584; and is the seq. because the intersection of {1,2,3,6,7,9} & {0, 4, 5, 8} = {}. Number of terms < 10^n: 2, 6, 10, 26, 41, 71, 84, 121, 129, 140, 141, 142. - Robert G. Wilson v, Apr 03 2009

H. Ibstedt, Solution to Problem 2623, "Exclusionary Powers", pp. 346-9, Journal of recreational Mathematics, Vol. 32 No. 4 2003-4 Baywood NY.

Robert G. Wilson v, Table of n, a(n) for n = 1..142 (full sequence)

Cf. A112322.

fQ[n_] := Intersection[ IntegerDigits[n], IntegerDigits[n^2]] == {}; Select[ Range@ 330, fQ@# &]^2 (* Robert G. Wilson v, Apr 03 2009 *)

Corrected and extended by Don Reble, Nov 22 2006

A254130 Numbers whose factorials are exclusionary: numbers n such that n and n! have no digits in common.

Original entry on oeis.org

0, 3, 5, 6, 7, 8, 9, 13, 16

Offset: 1

Felix Fröhlich, May 03 2015

Conjecture: The sequence is finite, with 16 being the last term.

If A182049 is finite, then this sequence is finite. If 41 is the largest term in A182049 (as is conjectured), then 16 is the largest term of this sequence. - M. F. Hasler, May 04 2015

			13! = 6227020800. 13 and 6227020800 have no digits in common, so 13 is a term of the sequence.

Cf. A000142, A112736, A112994, A111116.

Select[Range[0,16],DisjointQ[IntegerDigits[#],IntegerDigits[#!]]&] (* Ivan N. Ianakiev, May 04 2015 *)

is(n)=#setintersect(Set(digits(n)), Set(digits(n!)))==0

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

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A059930 Numbers n such that n and n^2 combined use different digits.

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A113951 Largest number whose n-th power is exclusionary (or 0 if no such number exists).

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A112735 Exclusionary squares.

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A254130 Numbers whose factorials are exclusionary: numbers n such that n and n! have no digits in common.

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