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%I #36 May 14 2020 07:41:17
%S 0,1,153,370,371,407
%N Fixed points for operation of repeatedly replacing a number with the sum of the cubes of its digits.
%C 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]
%C This is row n=3 of A252648. - _M. F. Hasler_, Apr 12 2015
%D J.-M. De Koninck, Ces nombres qui nous fascinent, Entry 153, p. 50, Ellipses, Paris 2008.
%D G. H. Hardy, A Mathematician's Apology, Cambridge, 1967.
%D 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.
%H H. Lehning, <a href="https://www.lehning.eu/uploads/1/2/0/8/120804163/articletangente_108_janvier_2006_la_migration_des_nombres_vers_le_bonheur.pdf">La migration des nombres vers le bonheur</a>, Tangente: L'aventure mathématique, pp. 27 No. 108 Jan-Feb 2006 Pole Paris.
%F A055012(a(n))=a(n); A165331(a(n))=0; subset of A031179. - _Reinhard Zumkeller_, Sep 17 2009
%e 1^3 + 5^3 + 3^3 = 153. 3^3+7^3 +0^3 = 370.
%o (PARI) for(n=0,10^5,A055012(n)==n&&print1(n",")) \\ _M. F. Hasler_, Apr 12 2015
%Y Cf. A005188, A023052, A046156.
%Y Cf. A165330, A035504, A008585, A165333, A165334, A165335.
%Y Cf. A052455, A052464, A124068, A124069, A226970, A003321.
%K nonn,fini,full,base
%O 1,3
%A Richard C. Schroeppel
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