

A061209


Numbers which are the cubes of their digit sum.


12




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^(d1) <= 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^(d1) 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 mth power whose digit sum is the mth 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



