A112994 -id:A112994 - cp's OEIS frontend

A112993 Exclusionary cubes: cubes of the terms in A112994.

8, 27, 343, 512, 19683, 79507, 103823, 110592, 140608, 148877, 250047, 314432, 778688, 3869893, 5088448, 6539203, 7077888, 18191447, 54010152, 67917312, 75686967, 96071912, 102503232, 109215352, 115501303, 146363183, 202262003, 224755712

Offset: 1

Lekraj Beedassy, Oct 13 2005

b-file is 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.

N. J. A. Sloane, Table of n, a(n) for n = 1..42 [From the Clifford Pickover link. Conjectured to be the full list of terms.]
Clifford A. Pickover, Extreme Challenges in Mathematics and Morals

Cf. A112994.

def ok(n):
    s = str(n)
    return len(s) == len(set(s)) and set(s) & set(str(n**3)) == set()
print([k**3 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**3)
    return alst
print(afull()) # Michael S. Branicky, Aug 27 2021

Corrected by N. J. A. Sloane, May 22 2008

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

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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A112993 Exclusionary cubes: cubes of the terms in A112994.

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

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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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Author

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Examples

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