A061209 Numbers which are the cubes of their digit sum.

0, 1, 512, 4913, 5832, 17576, 19683

Offset

1,3

Comments

It can be shown that 19683 = (1 + 9 + 6 + 8 + 3)^3 = 27^3 is the largest such number.

Numbers of Dudeney. - Philippe Deléham, May 11 2013

If a number n has d digits, 10^(d-1) <= n < 10^d, the cube of the digit sum is at most (d*9)^3 = 729*d^3; if d > 6 this is strictly smaller than 10^(d-1) and cannot be equal to n. See also A061211. - M. F. Hasler, Apr 12 2015

References

H. E. Dudeney, 536 Puzzles & Curious Problems, Souvenir Press, London, 1966, p. 36, #120.

Amarnath Murthy, The largest and the smallest m-th power whose digit sum is the m-th root. (To be published)

Links

Table of n, a(n) for n=1..7.

Henk Koppelaar and Peyman Nasehpour, On Hardy's Apology Numbers, arXiv:2008.08187 [math.NT], 2020.

Wikipedia, Dudeney number

Formula

a(n) = A007953(a(n))^3. - M. F. Hasler, Apr 12 2015

Example

4913 = (4 + 9 + 1 + 3)^3.

Mathematica

Select[Range[20000], Total[IntegerDigits[#]]^3==#&] (* Harvey P. Dale, Apr 11 2015 *)

Prog

(PARI) for(n=0, 999999, sumdigits(n)^3==n&&print1(n", ")) \\ M. F. Hasler, Apr 12 2015

Crossrefs

Cf. A007953, A061210, A061211, A252648.

Sequence in context: A186845 A239917 A114287 * A017259 A017367 A254899

Adjacent sequences: A061206 A061207 A061208 * A061210 A061211 A061212

Keyword

nonn,fini,full,base

Author

Amarnath Murthy, Apr 21 2001

Extensions

Initial term 0 added by M. F. Hasler, Apr 12 2015

Status

approved