Sequence of Dudeney Numbers

Theorem

The only Dudeney numbers are:

$0, 1, 8, 17, 18, 26, 27$

two of which are themselves cubes, and one of which is prime.

This sequence is A046459 in the On-Line Encyclopedia of Integer Sequences (N. J. A. Sloane (Ed.), 2008).

Proof

We have trivially that:

$\ds 0^3$

$=$

$\ds 0$

$\ds 1^3$

$=$

$\ds 1$

Then:

$\ds 8^3$

$=$

$\ds 512$

$\ds 8$

$=$

$\ds 5 + 1 + 2$

$\ds 17^3$

$=$

$\ds 4913$

$\ds 17$

$=$

$\ds 4 + 9 + 1 + 3$

$\ds 18^3$

$=$

$\ds 5832$

$\ds 18$

$=$

$\ds 5 + 8 + 3 + 2$

$\ds 26^3$

$=$

$\ds 17576$

$\ds 26$

$=$

$\ds 1 + 7 + 5 + 7 + 6$

$\ds 27^3$

$=$

$\ds 19683$

$\ds 27$

$=$

$\ds 1 + 9 + 6 + 8 + 3$

A quick empirical test shows that when $n = 46$, it is already too large to be the sum of the digits of its cube.

For $46 < n \le 54$, $n^3 \le 54^3 < 200 \, 000$.

Hence the sum of the digits of $n^3$ is less than:

$1 + 5 \times 9 = 46 < n$

For $54 < n < 100$, $n^3 < 10^6$.

Hence the sum of the digits of $n^3$ is less than:

$6 \times 9 = 54 < n$

For $n \ge 100$, let $n$ be a $d$-digit number, where $d \ge 3$.

Then $10^{d - 1} \le n < 10^d$ and $n^3 < 10^{3 d}$.

Hence the sum of the digits of $n^3$ is less than:

$\ds 3 d \times 9$

$=$

$\ds 27 d$

$\ds $

$\le$

$\ds 27 d + 63 d - 189$

$\ds $

$<$

$\ds 90 d - 170$

$\ds $

$=$

$\ds 10 \paren {1 + 9 \paren {d - 2} }$

$\ds $

$\le$

$\ds 10 \times \paren {1 + 9}^{d - 2}$

Bernoulli's Inequality

$\ds $

$\le$

$\ds n$

so no numbers greater than $46$ can have this property.

$\blacksquare$

Also reported as

Some sources (either deliberately or by oversight) do not include $0$ in this list.

Also see

Definition:Armstrong Number, with which the numbers in this entry appear frequently to be conflated

Historical Note

The earliest known appearance of this result is from Claude Séraphin Moret-Blanc in $1879$, although exactly where this was published is still to be identified.

Henry Ernest Dudeney subsequently published it in one of his own collections.

As a result, a number which equals the sum of the digits of its cube is now called a Dudeney number.

It continues to crop up occasionally in publications devoted to recreational mathematics.

Sources

1926: Henry Ernest Dudeney: Modern Puzzles ... (previous) ... (next): Solutions: $69$. -- Root Extraction
1968: Henry Ernest Dudeney: 536 Puzzles & Curious Problems ... (previous) ... (next): Answers: $120$. Root Extraction

1979: S.P. Mohanty and Hemant Kumar: Powers of Sums of Digits (Math. Mag. Vol. 52: pp. 310 – 312) www.jstor.org/stable/2689785

1983: François Le Lionnais and Jean Brette: Les Nombres Remarquables: $27$

1986: David Wells: Curious and Interesting Numbers ... (previous) ... (next): $17$
1986: David Wells: Curious and Interesting Numbers ... (previous) ... (next): $18$
1986: David Wells: Curious and Interesting Numbers ... (previous) ... (next): $26$
1986: David Wells: Curious and Interesting Numbers ... (previous) ... (next): $27$
1986: David Wells: Curious and Interesting Numbers ... (previous) ... (next): $4913$

1992: Joe Roberts: Lure of the Integers: $27$

1993: Monte James Zerger: The 'Number of Mathematics' (Journal of Recreational Mathematics Vol. 25, no. 4: pp. 247 – 251)

1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $17$
1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $18$
1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $26$
1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $27$
1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $4913$

Weisstein, Eric W. "Cubic Number." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/CubicNumber.html