A046197 Fixed points for operation of repeatedly replacing a number with the sum of the cubes of its digits.

0, 1, 153, 370, 371, 407

Offset: 1

Author

Richard C. Schroeppel

Keywords

• nonn
• fini
• full
• base

Comments

Suppose n has d digits; then the sum of the cubes of its digits is <= 729d and n >= 10^(d-1). So d <= 5. It is now easy to check that the numbers shown are the only solutions. [Corrected by M. F. Hasler, Apr 12 2015]

This is row n=3 of A252648. - M. F. Hasler, Apr 12 2015

Examples

			1^3 + 5^3 + 3^3 = 153. 3^3+7^3 +0^3 = 370.
		

References

• J.-M. De Koninck, Ces nombres qui nous fascinent, Entry 153, p. 50, Ellipses, Paris 2008.
• G. H. Hardy, A Mathematician's Apology, Cambridge, 1967.
• Alfred S. Posamentier, Math Charmers, Tantalizing Tidbits for the Mind, Prometheus Books, NY, 2003, pages 60-62.
• J. Shallit, Number theory and formal languages, in Emerging applications of number theory (Minneapolis, MN, 1996), 547-570, IMA Vol. Math. Appl., 109, Springer, New York, 1999.
• David Wells, The Penguin Dictionary of Curious and Interesting Numbers. Penguin Books, NY, 1986, Revised edition 1987. See p. 140.

Links

• H. Lehning, La migration des nombres vers le bonheur, Tangente: L'aventure mathématique, pp. 27 No. 108 Jan-Feb 2006 Pole Paris.

Crossrefs

Cf. A005188, A023052, A046156.

Cf. A165330, A035504, A008585, A165333, A165334, A165335.

Cf. A052455, A052464, A124068, A124069, A226970, A003321.

Programs

Mathematica

Select[Range[0,407],Total[IntegerDigits[#]^3]==# &] (* Stefano Spezia, Sep 08 2024 *)

PARI

for(n=0,10^5,A055012(n)==n&&print1(n",")) \\ M. F. Hasler, Apr 12 2015

Formula

A055012(a(n))=a(n); A165331(a(n))=0; subset of A031179. - Reinhard Zumkeller, Sep 17 2009