A111116 -id:A111116 - 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

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

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

A253574 Primes p such that digits of p do not appear in p^4.

Original entry on oeis.org

2, 3, 7, 53, 59, 67, 89, 383, 887, 2027, 3253, 5669, 7993, 8009, 9059, 53633, 54667, 56533, 88883, 272777777, 299222299, 383833883, 797769997

Offset: 1

Vincenzo Librandi, Jan 04 2015

Primes in A111116.
No further terms up to 10^9. - Felix Fröhlich, Jan 04 2015
No further terms up to 10^10. - Chai Wah Wu, Jan 06 2015
No further terms up to 2.5*10^13 - Giovanni Resta, Jun 01 2015
No further terms up to 10^19 (via A111116). - Michael S. Branicky, Jan 05 2022

			2 and 2^4=16 have no digits in common, hence 2 is in the sequence.

Cf. A111116.

Cf. primes such that digits of p do not appear in p^k: A030086 (k=2), A030087 (k=3), this sequence (k=4), no terms (k=5), A253575 (k=6), A253576 (k=7), A253577 (k=8), no terms (k=9), A253578 (k=10).

Select[Prime[Range[1000000]], Intersection[IntegerDigits[#], IntegerDigits[#^4]]=={} &]

forprime(p=1, 1e9, dip=digits(p); dipf=digits(p^4); sharedi=0; for(i=1, #dip, for(j=1, #dipf, if(dip[i]==dipf[j], sharedi++; break({2})))); if(sharedi==0, print1(p, ", "))) \\ Felix Fröhlich, Jan 04 2015

from sympy import isprime
A253574_list = [n for n in range(1,10**6) if set(str(n)) & set(str(n**4)) == set() and isprime(n)]
# Chai Wah Wu, Jan 06 2015

a(20)-a(23) from Felix Fröhlich, Jan 04 2015

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

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

A113318 Numbers whose biquadrates (fourth powers) are exclusionary.

Original entry on oeis.org

2, 3, 4, 7, 8, 9, 24, 27, 28, 32, 42, 52, 53, 58, 59, 67, 89, 93, 203, 258, 284, 324, 329, 832, 843, 2673

Offset: 1

Lekraj Beedassy, Oct 26 2005

The number m with no repeated digits has an exclusionary fourth power m^4 if the latter is made up of digits not appearing in m. Is a subsequence of A111116. Conjectured to be complete. For the corresponding exclusionary biquadrates m^4, see A113317.

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

ebQ[n_]:=Max[DigitCount[n]]==1&&Intersection[IntegerDigits[n], IntegerDigits[ n^4]]=={}; Select[Range[3000],ebQ] (* Harvey P. Dale, Aug 21 2013 *)

A253606 Numbers n such that digits of n are not present in n^8.

Original entry on oeis.org

3, 4, 8, 9, 22, 33, 43, 54, 59, 73, 222, 233, 353, 712, 777, 22224

Offset: 1

Chai Wah Wu, Jan 05 2015

a(17) > 10^9.

			4^8 = 65536 which does not contain the digit 4.

Cf. A111116, A253577.

a253606[n_] := Block[{f},
  f[x_] := MemberQ[IntegerDigits[x^8], #] & /@ IntegerDigits[x];
Select[Range@n, DeleteDuplicates@f[#] == {False} &]]; a253606[10^5] (* Michael De Vlieger, Jan 06 2015 *)

A253606_list = [n for n in range(1,10**6) if set(str(n)) & set(str(n**8)) == set()]

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

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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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A253574 Primes p such that digits of p do not appear in p^4.

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

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Mathematica

Extensions

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

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