cp's OEIS Frontend

A055012 Sum of cubes of the digits of n written in base 10.

%I A055012 #54 Jan 05 2025 01:29:43
%S A055012 0,1,8,27,64,125,216,343,512,729,1,2,9,28,65,126,217,344,513,730,8,9,
%T A055012 16,35,72,133,224,351,520,737,27,28,35,54,91,152,243,370,539,756,64,
%U A055012 65,72,91,128,189,280,407,576,793,125,126,133,152,189,250,341,468,637,854
%N A055012 Sum of cubes of the digits of n written in base 10.
%C A055012 For n > 1999, a(n) < n. The iteration of this map on n either stops at a fixed point (A046197) or has a period of 2 or 3: {55,250,133}, {136,244}, {160,217,352}, or {919,1459}. - _T. D. Noe_, Jul 17 2007
%C A055012 A165330 and A165331 give the final value and the number of steps when iterating until a fixed point or cycle is reached. - _Reinhard Zumkeller_, Sep 17 2009
%H A055012 T. D. Noe, <a href="/A055012/b055012.txt">Table of n, a(n) for n = 0..11000</a>
%H A055012 K. Iséki, <a href="http://dx.doi.org/10.3792/pja/1195523903">A problem of number theory</a>, Proc. Japan Academy 36 (1960), 578-583.
%H A055012 B. M. Stewart, <a href="http://dx.doi.org/10.4153/CJM-1960-032-9">Sums of functions of digits</a>, Canad. J. Math., 12 (1960), 374-389.
%H A055012 <a href="/index/Coi#Colombian">Index entries for Colombian or self numbers and related sequences</a>
%F A055012 a(n) = Sum_{k>=1} (floor(n/10^k) - 10*floor(n/10^(k+1)))^3. - _Hieronymus Fischer_, Jun 25 2007
%F A055012 a(10n+k) = a(n) + k^3, 0 <= k < 10. - _Hieronymus Fischer_, Jun 25 2007
%F A055012 From _Reinhard Zumkeller_, Sep 17 2009: (Start)
%F A055012 a(n) <= 729*A055642(n);
%F A055012 a(A165370(n)) = n and a(m) <> n for m < A165370(n);
%F A055012 a(A031179(n)) = A031179(n);
%F A055012 a(a(A165336(n))) = A165336(n) or a(a(a(A165336(n)))) = A165336(n). (End)
%F A055012 G.f. g(x) = Sum_{k>=0} (1-x^(10^k))*(x^(10^k)+8*x^(2*10^k)+27*x^(3*10^k)+64*x^(4*10^k)+125*x^(5*10^k)+216*x^(6*10^k)+343*x^(7*10^k)+512*x^(8*10^k)+729*x^(9*10^k))/((1-x)*(1-x^(10^(k+1))))
%F A055012 satisfies
%F A055012 g(x) = (x+8*x^2+27*x^3+64*x^4+125*x^5+216*x^6+343*x^7+512*x^8+729*x^9)/(1-x^10) + (1-x^10)*g(x^10)/(1-x). - _Robert Israel_, Jan 26 2017
%p A055012 A055012 := proc(n)
%p A055012         add(d^3,d=convert(n,base,10)) ;
%p A055012 end proc: # _R. J. Mathar_, Dec 15 2011
%t A055012 Total/@((IntegerDigits/@Range[0,60])^3) (* _Harvey P. Dale_, Jan 27 2012 *)
%t A055012 Table[Sum[DigitCount[n][[i]] i^3, {i, 9}], {n, 0, 60}] (* _Bruno Berselli_, Feb 01 2013 *)
%o A055012 (Magma) [0] cat [&+[d^3: d in Intseq(n)]: n in [1..60]]; // _Bruno Berselli_, Feb 01 2013
%o A055012 (PARI) A055012(n)=sum(i=1,#n=digits(n),n[i]^3) \\ _Charles R Greathouse IV_, Jul 01 2013
%o A055012 (Python)
%o A055012 def a(n): return sum(map(lambda x: x*x*x, map(int, str(n))))
%o A055012 print([a(n) for n in range(60)]) # _Michael S. Branicky_, Jul 13 2022
%Y A055012 Cf. A003132, A031179.
%Y A055012 Cf. A046197 Fixed points; A046459: integers equal to the sum of the digits of their cubes; A072884: 3rd-order digital invariants: the sum of the cubes of the digits of n equals some number k and the sum of the cubes of the digits of k equals n; A164883: cubes with the property that the sum of the cubes of the digits is also a cube.
%Y A055012 Cf. A003132, A007953, A055017, A076313, A076314, A165340.
%K A055012 base,nonn,easy,look
%O A055012 0,3
%A A055012 _Henry Bottomley_, May 31 2000
%E A055012 Edited by _M. F. Hasler_, Apr 12 2015
%E A055012 Iséki and Stewart links added by _Don Knuth_, Sep 07 2015