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.

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

Author

Lekraj Beedassy, Oct 13 2005

Keywords

• nonn
• base
• fini
• full

Comments

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

References

• 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.

Links

• Clifford A. Pickover, Extreme Challenges in Mathematics and Morals

Crossrefs

Subsequence of A029785.

The corresponding cubes are in A112993.

Programs

Mathematica

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

PARI

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

Python

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

Python

# 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

Extensions

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

Definition clarified by Harvey P. Dale, Sep 06 2021