A029785 -id:A029785 - 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

A113316 Fourth powers m^4 none of whose digits are present in their corresponding roots m.

Original entry on oeis.org

16, 81, 256, 2401, 4096, 6561, 331776, 531441, 614656, 1048576, 1185921, 3111696, 7311616, 7890481, 11316496, 12117361, 20151121, 35153041, 59969536, 62742241, 74805201, 1664966416, 1698181681, 4430766096, 6505390336, 8428892481

Offset: 1

Lekraj Beedassy, Oct 26 2005

For the associated root m, see A111116.

Giovanni Resta, Table of n, a(n) for n = 1..179 (terms < 10^76, first 100 terms from Harvey P. Dale)

Cf. A111116, A029783, A029784, A029785, A029786.

Select[Range[350]^4,Intersection[IntegerDigits[#], IntegerDigits[ Surd[ #,4]]] =={}&] (* Harvey P. Dale, Dec 19 2015 *)

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

A030087 Primes such that digits of p do not appear in p^3.

Original entry on oeis.org

2, 3, 7, 43, 47, 53, 157, 223, 263, 487, 577, 587, 823, 4657, 5657, 6653, 7177, 8287, 9343, 26777, 36293, 46477, 58787, 72727, 75707, 176777, 363313, 530353, 566653, 959953, 1771787, 2525557, 2555353, 2626277, 3656363, 4414447, 7110707, 8448343, 20700077, 54475457, 71117177, 72722977, 135135113, 393321293, 457887457, 505053053, 672722627

Offset: 1

Patrick De Geest, Dec 11 1999

Primes of sequence A029785. - Michel Marcus, Jan 04 2015

			2 and 2^3=8 have no digits in common, hence 2 is in the sequence.

Michael S. Branicky, Table of n, a(n) for n = 1..86 (terms < 10^19; terms 74..86 via A029785)

Cf. A029785 (digits of n are not present in n^3), A030086 (similar, with p^2), A253574 (similar, with p^4).

Select[Prime[Range[1500000]], Intersection[IntegerDigits[#], IntegerDigits[#^3]]=={} &] (* Vincenzo Librandi, Jan 04 2015 *)

lista(nn) = {forprime (n=1, nn, if (#setintersect(Set(vecsort(digits(n^3))), Set(vecsort(digits(n)))) == 0, print1(n, ", ")); ); } \\ Michel Marcus, Jan 04 2015

from sympy import isprime
A030087_list = [n for n in range(1,10**6) if set(str(n)) & set(str(n**3)) == set() and isprime(n)]
# Chai Wah Wu, Jan 05 2015

Changed offset from 0 to 1 and more terms from Vincenzo Librandi, Jan 04 2015

a(40)-a(47) from Chai Wah Wu, Jan 05 2015

A112994 Numbers whose cubes are exclusionary: numbers k such that k has no repeated digits and k and k^3 have no digits in common.

Original entry on oeis.org

2, 3, 7, 8, 27, 43, 47, 48, 52, 53, 63, 68, 92, 157, 172, 187, 192, 263, 378, 408, 423, 458, 468, 478, 487, 527, 587, 608, 648, 692, 823, 843, 918, 1457, 1587, 1592, 4657, 4732, 5692, 6058, 6378, 7658

Offset: 1

Lekraj Beedassy, Oct 13 2005

A number k with no repeated digits has an exclusionary cube k^3 if the latter is made up of digits not appearing in k. (This is a subsequence of A029785.) For the corresponding exclusionary cubes see A112993. Conjectured to be complete.

Data are complete: there are 42 terms. - Michael S. Branicky, Aug 27 2021

H. Ibstedt, Solution to Problem 2623, "Exclusionary Powers", pp. 346-9, Journal of Recreational Mathematics, vol. 32 No.4 2003-4, Baywood NY.
Clifford A. Pickover, A Passion for Mathematics, Wiley, 2005; see p. 60.

Clifford A. Pickover, Extreme Challenges in Mathematics and Morals

Subsequence of A029785.

The corresponding cubes are in A112993.

Select[Range[8000],Max[DigitCount[#]]==1&&Intersection[IntegerDigits[ #],IntegerDigits[#^3]]=={}&] (* Harvey P. Dale, Sep 06 2021 *)

isok(n) = my(digs = digits(n)); (#digs == #Set(digs)) && (#setintersect(Set(digs), Set(digits(n^3))) == 0); \\ Michel Marcus, Oct 26 2013

def ok(n):
    s = str(n)
    return len(s) == len(set(s)) and set(s) & set(str(n**3)) == set()
print([k for k in range(7659) if ok(k)]) # Michael S. Branicky, Aug 27 2021

# version for verifying full sequence
from itertools import 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 afull():
    alst = []
    for k in no_repeated_digits():
        if set(str(k)) & set(str(k**3)) == set():
            alst.append(k)
    return alst
print(afull()) # Michael S. Branicky, Aug 27 2021

Missing term 468 added by N. J. A. Sloane, May 22 2008

Definition clarified by Harvey P. Dale, Sep 06 2021

A247883 Consecutive exclusionary cubes: Digits of n are not present in n^3 and digits of n+1 are not present in (n+1)^3.

Original entry on oeis.org

2, 7, 47, 52, 187, 222, 477, 587, 5522, 6777

Offset: 1

Derek Orr, Sep 25 2014

If it exists, a(11) > 10^7.

All terms == 2 (mod 5). - Robert Israel, Mar 10 2025

Cf. A029785.

filter:= proc(n) convert(convert(n,base,10),set) intersect convert(convert(n^3,base,10),set) = {} end proc:
select(t -> filter(t) and filter(t+1), [seq(i,i=2..10^6, 5)]); # Robert Israel, Mar 10 2025

for(n=1,10^6,s=digits(n);t=digits(n+1);s3=digits(n^3);t3=digits((n+1)^3);if(#vecsort(concat(s,s3),,8)==#vecsort(s,,8)+#vecsort(s3,,8)&&#vecsort(concat(t,t3),,8)==#vecsort(t,,8)+#vecsort(t3,,8),print1(n,", ")))

for n in range(10**6):
  s, t = str(n), str(n+1)
  s3, t3 = str(n**3), str((n+1)**3)
  c = 0
  for i in s:
    if s3.count(i):
      c += 1
      break
  for j in t:
    if t3.count(j):
      c += 1
      break
  if not c:
    print(n, end=', ')

Definition corrected by Robert Israel, Mar 10 2025

cp's OEIS Frontend

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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A113316 Fourth powers m^4 none of whose digits are present in their corresponding roots m.

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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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A030087 Primes such that digits of p do not appear in p^3.

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A112994 Numbers whose cubes are exclusionary: numbers k such that k has no repeated digits and k and k^3 have no digits in common.

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