A113316 -id:A113316 - 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

A111116 Numbers n such that digits of n are not present in n^4.

Original entry on oeis.org

2, 3, 4, 7, 8, 9, 24, 27, 28, 32, 33, 42, 52, 53, 58, 59, 67, 77, 88, 89, 93, 202, 203, 258, 284, 303, 324, 329, 377, 383, 422, 669, 818, 832, 843, 878, 882, 887, 949, 2027, 2042, 2673, 3144, 3222, 3253, 3302, 3308, 3737, 3773, 3953, 3979, 3983, 4779, 5353, 5669

Offset: 1

Lekraj Beedassy, Oct 15 2005

The number of k-digit numbers for which this occurs is: 6,15,18,32,21,14,20,...

Giovanni Resta, Table of n, a(n) for n = 1..179 (terms < 10^19, first 136 terms from Chai Wah Wu)

Cf. A029783, A029785.

For the corresponding n^4, see A113316.

Select[Range[6000], Intersection[IntegerDigits[ # ], IntegerDigits[ #^4]] == {} &] (* Ray Chandler, Oct 17 2005 *)
fQ[n_] := Intersection[ Union[ IntegerDigits[n]], Union[ IntegerDigits[n^4]]] == {}; Select[ Range[ 5887], fQ[ # ] &] (* Robert G. Wilson v *)

A111116_list = [n for n in range(1,10**6) if set(str(n)) & set(str(n**4)) == set()]
# Chai Wah Wu, Jan 05 2015

Corrected and extended by Robert G. Wilson v and Ray Chandler, Oct 17 2005

A029785 Numbers k whose cube k^3 has no digit in common with k.

Original entry on oeis.org

2, 3, 7, 8, 22, 27, 43, 47, 48, 52, 53, 63, 68, 77, 92, 157, 172, 177, 187, 188, 192, 222, 223, 252, 263, 303, 378, 408, 423, 442, 458, 468, 477, 478, 487, 527, 552, 558, 577, 587, 588, 608, 648, 692, 707, 772, 808, 818, 823, 843, 888, 918, 922

Offset: 1

Patrick De Geest

Original name: Digits of n are not present in n^3.

Might be called "Exclusionary Cubes", although this might be reserved for terms having no duplicate digits, cf. link to rec.puzzles discussion group. In that case the largest term would be 7658 = A113951(3). - M. F. Hasler, Oct 17 2018; corrected thanks to David Radcliffe and Michel Marcus, Apr 30 2020

			k = 80800000008880080808880080088 is in the sequence because the 87-digit number k^3 has only digits 1, ..., 7 and 9. - _M. F. Hasler_, Oct 16 2018

Giovanni Resta, Table of n, a(n) for n = 1..1699 (terms < 10^19, first 528 terms from Charles R Greathouse IV)
Cliff Pickover et al, Exclusionary Squares and Cubes, rec.puzzles topic on google groups, January 2002

Cf. A029786, A029783, A029784, A111116, A113316.

Select[Range[5000], Intersection[IntegerDigits[#], IntegerDigits[#^3]]=={}&] (* Vincenzo Librandi, Oct 04 2013 *)

is(n)=my(d=Set(digits(n))); setminus(d,Set(digits(n^3)))==d \\ Charles R Greathouse IV, Oct 02 2013

is_A029785(n)=setintersect(Set(digits(n)),Set(digits(n^3)))==[] \\ M. F. Hasler, Oct 16 2018

Name reworded by Jon E. Schoenfield and M. F. Hasler, Oct 16 2018

A029784 Squares such that digits of sqrt(n) are not present in n.

Original entry on oeis.org

4, 9, 16, 49, 64, 81, 289, 324, 484, 576, 841, 1089, 1156, 1444, 1521, 1936, 2209, 2809, 2916, 3249, 3364, 3481, 3844, 4489, 5184, 5929, 6241, 7056, 7744, 8464, 8836, 20736, 24649, 24964, 29929, 34969, 35344, 36864, 37636, 43681, 44944, 50176, 56169, 56644

Offset: 1

Patrick De Geest

Giovanni Resta, Table of n, a(n) for n = 1..10000

Cf. A029783, A029785, A029786, A111116, A113316.

Select[Range[500], {} == Intersection @@ IntegerDigits[{#, #^2}] &]^2 (* Giovanni Resta, Aug 08 2018 *)

a(n) = A029783(n)^2. - Michel Marcus, Aug 08 2018

More terms from Giovanni Resta, Aug 08 2018

A029786 Cubes such that digits of cube root of n are not present in n.

Original entry on oeis.org

8, 27, 343, 512, 10648, 19683, 79507, 103823, 110592, 140608, 148877, 250047, 314432, 456533, 778688, 3869893, 5088448, 5545233, 6539203, 6644672, 7077888, 10941048, 11089567, 16003008, 18191447, 27818127, 54010152, 67917312, 75686967, 86350888, 96071912

Offset: 1

Patrick De Geest

Giovanni Resta, Table of n, a(n) for n = 1..1699 (terms < 10^57)

Cf. A029785, A029783, A029784, A111116, A113316.

Select[Range[1000], {} == Intersection @@ IntegerDigits[{#, #^3}] &]^3 (* Giovanni Resta, Aug 08 2018 *)

More terms from Giovanni Resta, Aug 08 2018

A113317 Exclusionary biquadrates (fourth powers).

Original entry on oeis.org

16, 81, 256, 2401, 4096, 6561, 331776, 531441, 614656, 1048576, 3111696, 7311616, 7890481, 11316496, 12117361, 21051121, 62742241, 74805201, 1698181681, 4430766096, 6505390336, 11019960576, 11716114081, 479174066176, 505022001201, 51050010415041

Offset: 1

Lekraj Beedassy, Oct 26 2005

An exclusionary biquadrate m^4 is one sharing no digit in common with its root m made up of distinct digits.

			331776 = 24^4 is in the sequence as 331776 and 24 have no digits in common and 24 has distinct digits. - _David A. Corneth_, May 12 2025

Shyam Sunder Gupta, "Exploring the Beauty of Fascinating Numbers", Springer, pp. 43-44.
H. Ibstedt, Solution to Problem 2623, "Exclusionary Powers", pp. 346-9, Journal of Recreational Mathematics, vol. 32 No.4 2003-4 Baywood NY.

Patrick De Geest, Reference table for squares, cubes and biquadrates

Subsequence of A000583. Subsequence of A113316.

Roots are given by A113318.

a(n) = A113318(n)^4.

Corrected by Don Reble, Nov 22 2006

a(25)-a(26) added by Patrick De Geest, May 12 2025

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

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A111116 Numbers n such that digits of n are not present in n^4.

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A029785 Numbers k whose cube k^3 has no digit in common with k.

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A029784 Squares such that digits of sqrt(n) are not present in n.

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A029786 Cubes such that digits of cube root of n are not present in n.

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A113317 Exclusionary biquadrates (fourth powers).

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