A165331 -id:A165331 - cp's OEIS frontend

A165340 Triangle read by rows: T(n,0) = smallest number m such that A165331(m)=n and A165330(m)=153; T(n,k+1) = sum of cubes of digits of T(n,k), 0<=k

153, 135, 153, 18, 513, 153, 3, 27, 351, 153, 9, 729, 1080, 513, 153, 12, 9, 729, 1080, 513, 153, 33, 54, 189, 1242, 81, 513, 153, 114, 66, 432, 99, 1458, 702, 351, 153, 78, 855, 762, 567, 684, 792, 1080, 513, 153, 126, 225, 141, 66, 432, 99, 1458, 702, 351

Offset: 0

Reinhard Zumkeller, Sep 17 2009

T(n,k+1) = A055012(T(n,k)), 0 <= k < n;
A165331(T(n,k)) = n - k;
A165330(T(n,k)) = 153; T(n,n) = 153;
10^10 < T(15,0) <= 22222599999999999999999,
T(14,0) = 12558 = A055012(22222599999999999999999).

			The triangle begins:
n=0: 153,
n=1: 135 -> 1+3^3+5^3=153,
n=2: 18 -> 1+8^3=513 -> 5^3+1+3^3=153,
n=3: 3 -> 3^3=27 -> 2^3+7^3=351 -> 3^3+5^3+1=153,
n=4: 9 -> 9^3=729 -> 7^3+2^3+9^3=1080 -> 1+0+8^3+0=513 -> 5^3+1+3^3=153,
n=5: 12 -> 1+2^3=9 -> 9^3=729 -> 7^3+2^3+9^3=1080 -> 1+0+8^3+0=513 -> 5^3+1+3^3=153,
n=6: 33 -> 2*3^3=54 -> 5^3+4^3=189 -> 1+8^3+9^3=1242 -> 1+2^3+4^3+2^3=81 -> 8^3+1=513 -> 5^3+1+3^3=153.

R. Zumkeller, Rows 0 to 14 of the triangle, flattened.

A008585.

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

Original entry on oeis.org

0, 1, 8, 27, 64, 125, 216, 343, 512, 729, 1, 2, 9, 28, 65, 126, 217, 344, 513, 730, 8, 9, 16, 35, 72, 133, 224, 351, 520, 737, 27, 28, 35, 54, 91, 152, 243, 370, 539, 756, 64, 65, 72, 91, 128, 189, 280, 407, 576, 793, 125, 126, 133, 152, 189, 250, 341, 468, 637, 854

Offset: 0

Henry Bottomley, May 31 2000

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

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

T. D. Noe, Table of n, a(n) for n = 0..11000
K. Iséki, A problem of number theory, Proc. Japan Academy 36 (1960), 578-583.
B. M. Stewart, Sums of functions of digits, Canad. J. Math., 12 (1960), 374-389.
Index entries for Colombian or self numbers and related sequences

Cf. A003132, A031179.
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.
Cf. A003132, A007953, A055017, A076313, A076314, A165340.

[0] cat [&+[d^3: d in Intseq(n)]: n in [1..60]]; // Bruno Berselli, Feb 01 2013

A055012 := proc(n)
        add(d^3,d=convert(n,base,10)) ;
end proc: # R. J. Mathar, Dec 15 2011

Total/@((IntegerDigits/@Range[0,60])^3) (* Harvey P. Dale, Jan 27 2012 *)
Table[Sum[DigitCount[n][[i]] i^3, {i, 9}], {n, 0, 60}] (* Bruno Berselli, Feb 01 2013 *)

A055012(n)=sum(i=1,#n=digits(n),n[i]^3) \\ Charles R Greathouse IV, Jul 01 2013

def a(n): return sum(map(lambda x: x*x*x, map(int, str(n))))
print([a(n) for n in range(60)]) # Michael S. Branicky, Jul 13 2022

a(n) = Sum_{k>=1} (floor(n/10^k) - 10*floor(n/10^(k+1)))^3. - Hieronymus Fischer, Jun 25 2007
a(10n+k) = a(n) + k^3, 0 <= k < 10. - Hieronymus Fischer, Jun 25 2007
From Reinhard Zumkeller, Sep 17 2009: (Start)
a(n) <= 729*A055642(n);
a(A165370(n)) = n and a(m) <> n for m < A165370(n);
a(A031179(n)) = A031179(n);
a(a(A165336(n))) = A165336(n) or a(a(a(A165336(n)))) = A165336(n). (End)
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))))
satisfies
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

Edited by M. F. Hasler, Apr 12 2015

Iséki and Stewart links added by Don Knuth, Sep 07 2015

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

Original entry on oeis.org

0, 1, 153, 370, 371, 407

Offset: 1

Richard C. Schroeppel

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

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

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.

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

Cf. A005188, A023052, A046156.
Cf. A165330, A035504, A008585, A165333, A165334, A165335.
Cf. A052455, A052464, A124068, A124069, A226970, A003321.

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

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

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

A046156 Limit set for operation of repeatedly replacing a number with the sum of the cubes of its digits.

Original entry on oeis.org

0, 1, 55, 133, 136, 153, 160, 217, 244, 250, 352, 370, 371, 407, 919, 1459

Offset: 1

Richard C. Schroeppel

Range of A165330; A165330(a(n))=a(n); A165331(a(n))=0. - Reinhard Zumkeller, Sep 17 2009

Cf. A046197, A113708.

lst = {}; k = 0; While[k < 1500, a = NestWhile[Plus @@ (IntegerDigits@ #^3) &, k, Unequal, All]; If[FreeQ[lst, a], AppendTo[lst, a]]; k++]; Sort@ lst (* Robert G. Wilson v, Jan 19 2006, revised Jan 03 2015 *)
Table[Nest[Total[IntegerDigits[#]^3]&,n,30],{n,0,1500}]//Union (* Harvey P. Dale, Aug 04 2018 *)

A182111 Number of iterations of the map n -> sum of the cubes of the decimal digits of n.

Original entry on oeis.org

1, 7, 3, 6, 6, 10, 6, 6, 4, 1, 8, 5, 5, 6, 10, 3, 8, 2, 2, 7, 5, 4, 7, 3, 3, 8, 2, 4, 3, 3, 5, 7, 6, 3, 6, 6, 1, 8, 6, 6, 6, 3, 3, 7, 5, 5, 1, 6, 4, 6, 10, 3, 6, 5, 3, 5, 5, 8, 10, 10, 3, 8, 6, 5, 5, 6, 7, 11, 6, 6, 8, 2, 1, 1, 5, 7, 7, 8, 4, 6, 2, 4, 8, 6, 8

Offset: 1

Michel Lagneau, Apr 12 2012

a(n) is the number of times you obtain the sums of cubes of digits of n before reaching a fixed point (last number of the cycle).

			a(3) = 3 because :
3^3  = 27 -> 2^3 + 7^3 = 351;
351 -> 3^3 + 5^3 + 1^3 = 153;
153 -> 1^3+5^3+3^3 = 153 is the end because this number is already in the trajectory. Hence we obtain the map : 3 -> 27 -> 351 -> 153 with 3 iterations.

Cf. A000578, A055012, A152077, A160862, A165331.

a:= proc(n) local k, m, s; m:= n; s:= {};
      for k from 0 do
        m:= add(i^3, i=convert(m, base, 10));
        if m in s then return k fi;
        s:= s union {m}
      od
    end:
seq(a(n), n=1..85);  # Alois P. Heinz, Mar 01 2018

A003620 Number of iterations until 3n reaches 153 under x goes to sum of cubes of digits map.

Original entry on oeis.org

3, 10, 4, 5, 10, 2, 5, 3, 2, 3, 6, 6, 6, 3, 5, 6, 10, 5, 5, 10, 6, 6, 6, 2, 5, 8, 2, 6, 8, 4, 6, 6, 4, 5, 10, 2, 4, 7, 11, 5, 7, 9, 10, 7, 1, 6, 7, 11, 7, 10, 0, 6, 8, 9, 6, 4, 11, 7, 13, 2, 6, 4, 4, 10, 8, 4, 5, 3, 2, 5, 7, 9, 10, 4, 8, 8, 7, 5, 10, 3

Offset: 1

N. J. A. Sloane, Mira Bernstein

a(n) = A165331(A008585(n)). [From Reinhard Zumkeller, Nov 21 2009]

J. Roberts, Lure of the Integers, Math. Assoc. America, 1992, p. 13.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Cf. A055012. [From Reinhard Zumkeller, Nov 21 2009]

Table[Length[NestWhileList[Total[IntegerDigits[#]^3]&,3n,#!=153&]]-1,{n,80}] (* Harvey P. Dale, Dec 19 2011 *)

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A165340 Triangle read by rows: T(n,0) = smallest number m such that A165331(m)=n and A165330(m)=153; T(n,k+1) = sum of cubes of digits of T(n,k), 0<=k

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

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

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A046156 Limit set for operation of repeatedly replacing a number with the sum of the cubes of its digits.

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A182111 Number of iterations of the map n -> sum of the cubes of the decimal digits of n.

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A003620 Number of iterations until 3n reaches 153 under x goes to sum of cubes of digits map.

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