A165330 -id:A165330 - 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.

A008585 a(n) = 3*n.

Original entry on oeis.org

0, 3, 6, 9, 12, 15, 18, 21, 24, 27, 30, 33, 36, 39, 42, 45, 48, 51, 54, 57, 60, 63, 66, 69, 72, 75, 78, 81, 84, 87, 90, 93, 96, 99, 102, 105, 108, 111, 114, 117, 120, 123, 126, 129, 132, 135, 138, 141, 144, 147, 150, 153, 156, 159, 162, 165, 168, 171, 174, 177

Offset: 0

N. J. A. Sloane

If n != 1 and n^2+2 is prime then n is a member of this sequence. - Cino Hilliard, Mar 19 2007
Multiples of 3. Positive members of this sequence are the third transversal numbers (or 3-transversal numbers): Numbers of the 3rd column of positive numbers in the square array of nonnegative and polygonal numbers A139600. Also, numbers of the 3rd column in the square array A057145. - Omar E. Pol, May 02 2008
Numbers n for which polynomial 27*x^6-2^n is factorizable. - Artur Jasinski, Nov 01 2008
1/7 in base-2 notation = 0.001001001... = 1/2^3 + 1/2^6 + 1/2^9 + ... - Gary W. Adamson, Jan 24 2009
A165330(a(n)) = 153 for n > 0; subsequence of A031179. - Reinhard Zumkeller, Sep 17 2009
A011655(a(n)) = 0. - Reinhard Zumkeller, Nov 30 2009
A215879(a(n)) = 0. - Reinhard Zumkeller, Dec 28 2012
Moser conjectured, and Newman proved, that the terms of this sequence are more likely to have an even number of 1s in binary than an odd number. The excess is an undulating multiple of n^(log 3/log 4). See also Coquet, who refines this result. - Charles R Greathouse IV, Jul 17 2013
Integer areas of medial triangles of integer-sided triangles.
Also integer subset of A188158(n)/4.
A medial triangle MNO is formed by joining the midpoints of the sides of a triangle ABC. The area of a medial triangle is A/4 where A is the area of the initial triangle ABC. - Michel Lagneau, Oct 28 2013
From Derek Orr, Nov 22 2014: (Start)
Let b(0) = 0, and b(n) = the number of distinct terms in the set of pairwise sums {b(0), ... b(n-1)} + {b(0), ... b(n-1)}. Then b(n+1) = a(n), for n > 0.
Example: b(1) = the number of distinct sums of {0} + {0}. The only possible sum is {0} so b(1) = 1. b(2) = the number of distinct sums of {0,1} + {0,1}. The possible sums are {0,1,2} so b(2) = 3. b(3) = the number of distinct sums of {0,1,3} + {0,1,3}. The possible sums are {0, 1, 2, 3, 4, 6} so b(3) = 6. This continues and one can see that b(n+1) = a(n). (End)
Number of partitions of 6n into exactly 2 parts. - Colin Barker, Mar 23 2015
Partial sums are in A045943. - Guenther Schrack, May 18 2017
Number of edges in a maximal planar graph with n+2 vertices, n > 0 (see A008486 comments). - Jonathan Sondow, Mar 03 2018
Also numbers such that when the leftmost digit is moved to the unit's place the result is divisible by 3. - Stefano Spezia, Jul 08 2025

			G.f.: 3*x + 6*x^2 + 9*x^3 + 12*x^4 + 15*x^5 + 18*x^6 + 21*x^7 + ...

A. H. Beiler, Recreations in the Theory of Numbers, Dover, NY, 1964, p. 189.

Vincenzo Librandi, Table of n, a(n) for n = 0..5000
J. Coquet, A summation formula related to the binary digits, Inventiones Mathematicae 73 (1983), pp. 107-115.
Charles Cratty, Samuel Erickson, Frehiwet Negass, and Lara Pudwell, Pattern Avoidance in Double Lists, preprint, 2015.
A. S. Fraenkel, New games related to old and new sequences, INTEGERS, Electronic J. of Combinatorial Number Theory, Vol. 4, Paper G6, 2004.
John Graham-Cumming, The hollow triangular numbers are divisible by three (2013)
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 315
Tanya Khovanova, Recursive Sequences
D. J. Newman, On the number of binary digits in a multiple of three, Proc. Amer. Math. Soc. 21 (1969) 719-721.
Franck Ramaharo, Statistics on some classes of knot shadows, arXiv:1802.07701 [math.CO], 2018.
Luis Manuel Rivera, Integer sequences and k-commuting permutations, arXiv preprint arXiv:1406.3081 [math.CO], 2014-2015.
Wikipedia, Maximal planar graphs
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Row / column 3 of A004247 and of A325820.
Cf. A016957, A057145, A139600, A139606, A001651 (complement), A032031 (partial products), A190944 (binary), A061819 (base 4).
Cf. A031179, A008486, A008591, A017197, A017233, A045943.

a008585 = (* 3)
a008585_list = iterate (+ 3) 0  -- Reinhard Zumkeller, Feb 19 2013

[3*n: n in [0..60]]; // Vincenzo Librandi, Jul 23 2011

Range[0, 500, 3] (* Vladimir Joseph Stephan Orlovsky, May 26 2011 *)

makelist(3*n,n,0,30); /* Martin Ettl, Nov 12 2012 */

a(n)=3*n \\ Charles R Greathouse IV, Jun 28 2013

G.f.: 3*x/(1-x)^2. - R. J. Mathar, Oct 23 2008
a(n) = A008486(n), n > 0. - R. J. Mathar, Oct 28 2008
G.f.: A(x) - 1, where A(x) is the g.f. of A008486. - Gennady Eremin, Feb 20 2021
a(n) = Sum_{k=0..inf} A030308(n,k)*A007283(k). - Philippe Deléham, Oct 17 2011
E.g.f.: 3*x*exp(x). - Ilya Gutkovskiy, May 18 2016
From Guenther Schrack, May 18 2017: (Start)
a(3*k) = a(a(k)) = A008591(n).
a(3*k+1) = a(a(k) + 1) = a(A016777(n)) = A017197(n).
a(3*k+2) = a(a(k) + 2) = a(A016789(n)) = A017233(n). (End)

Partially edited by Joerg Arndt, Mar 11 2010

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

A035504 Numbers that eventually reach 1 under "x -> sum of cubes of digits of x".

Original entry on oeis.org

1, 10, 100, 112, 121, 211, 778, 787, 877, 1000, 1012, 1021, 1102, 1120, 1189, 1198, 1201, 1210, 1234, 1243, 1324, 1342, 1423, 1432, 1579, 1597, 1759, 1795, 1819, 1891, 1918, 1957, 1975, 1981, 2011, 2101, 2110, 2134, 2143, 2314, 2341, 2413, 2431, 2779

Offset: 1

N. J. A. Sloane

Subsequence of A016777; a(n) mod 3 = 1; A165330(a(n))=1. [Reinhard Zumkeller, Sep 17 2009]

R. K. Guy, Unsolved Problems Number Theory, Sect. E34.

R. Zumkeller, Table of n, a(n) for n = 1..1000
R. Styer, Smallest Examples of Strings of Consecutive Happy Numbers, J. Int. Seq. 13 (2010), 10.6.3, Section 4.

Cf. A007770.

Cf. A046197, A008585, A165333, A165334, A165335; subsequence of A031179.

f[n_]:=Plus@@(IntegerDigits[n]^3);Trajectory[n_] := Most[NestWhileList[f, n, UnsameQ, All]];Select[Range[2780],Last[Trajectory[#]]==1 &] (* Ant King, May 24 2013 *)

More terms from Larry Reeves (larryr(AT)acm.org), Mar 31 2000

A165331 Number of iterations until a fixed point or cycle is reached when repeatedly replacing a number with the sum of the cubes of its digits.

Original entry on oeis.org

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

Offset: 0

Reinhard Zumkeller, Sep 17 2009

a(A046197(k)) = 0 for k: 1 <= k <= 6;
a(A046156(k)) = 0 for k: 1 <= k <= 16;
a(A165330(n)) = 0;
a(A165340(n,k)) = n - k, 0<=k<=n.
a(A008585(n)) = A003620(n), n>0. [From Reinhard Zumkeller, Nov 21 2009]

R. Zumkeller, Table of n, a(n) for n = 0..20000

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 *)

A154877 Numbers that eventually reach the cycle 160-217-352 under "x -> sum of cubes of digits of x" (see A055012).

Original entry on oeis.org

16, 22, 61, 79, 97, 106, 115, 127, 151, 160, 172, 202, 217, 220, 229, 235, 238, 253, 271, 283, 292, 325, 328, 352, 382, 388, 445, 454, 457, 475, 511, 523, 532, 544, 547, 574, 601, 610, 709, 712, 721, 745, 754, 790, 823, 832, 838, 883, 907, 922, 970, 1006

Offset: 1

Avik Roy (avik_3.1416(AT)yahoo.co.in), Jan 16 2009

All the numbers are of the form 3n+1.
A165330(a(n)) = 160;
Subsequence of A165336.

			Taking 79 as an example; 7^3+9^3=1072, 1^3+0^3+7^3+2^3=352, 3^3+5^3+2^3=160, 1^3+6^3+0^3=217, 2^3+1^3+7^3=352.
a(15)=229: 229 -> 2*2^3+9^3=745 -> 7^3+4^3+5^3+1=532 -> 5^3+3^3+2^3=160 -> 217 -> 352 -> 160 ... .

Cf. A154820, A154877, A165330, A165336, A165337, A165339.

okQ[n_]:=MemberQ[NestList[Total[IntegerDigits[#]^3]&,n,20],160]; Select[Range[1200],okQ] (* Harvey P. Dale, Jun 20 2011 *)

Further terms added by Avik Roy (avik_3.1416(AT)yahoo.co.in), Jan 20 2009
Corrected by Reinhard Zumkeller, Sep 17 2009.
Confirmed by Harvey P. Dale, Jun 20 2011
Entry revised by N. J. A. Sloane, Oct 13 2018 (merging older duplicate entry with this one).

A165333 Numbers that eventually reach the fixed point 370 under "x -> sum of cubes of digits of x" (see A055012).

Original entry on oeis.org

7, 19, 34, 37, 43, 58, 67, 70, 73, 76, 85, 88, 91, 109, 118, 124, 139, 142, 145, 148, 154, 157, 166, 169, 175, 178, 181, 184, 187, 190, 193, 196, 214, 223, 226, 232, 241, 247, 259, 262, 268, 274, 277, 286, 295, 304, 307, 319, 322, 334, 340, 343, 346, 355, 358

Offset: 1

Reinhard Zumkeller, Sep 17 2009

A165330(a(n)) = 370;

Subsequence of A031179 and of A016777; a(n) mod 3 = 1.

			a(3)=34: 34 -> 3^3+4^3=91 -> 9^3+1=730 -> 7^3+3^3+0=370.

Cf. A031179, A046197, A035504, A008585, A165334, A165335.

f[n_] := Plus@@(IntegerDigits[n]^3); Trajectory[n_] := Most[NestWhileList[f, n, UnsameQ ,All]]; Select[Range[358], Last[Trajectory[#]] == 370&] (* Ant King, May 24 2013 *)

A165334 Numbers that eventually reach the fixed point 371 under "x -> sum of cubes of digits of x" (see A055012).

Original entry on oeis.org

2, 5, 8, 11, 14, 17, 20, 23, 26, 29, 32, 35, 38, 41, 44, 50, 53, 56, 59, 62, 65, 68, 71, 80, 83, 86, 92, 95, 101, 104, 107, 110, 113, 116, 119, 122, 125, 128, 131, 134, 137, 140, 143, 146, 149, 152, 155, 158, 161, 164, 167, 170, 173, 176, 179, 182, 185, 188, 191

Offset: 1

Reinhard Zumkeller, Sep 17 2009

A165330(a(n)) = 371;
Subsequence of A031179;
complement of A165335 with respect to A016789; a(n) mod 3 = 2.

			a(10)=29: 29 -> 2^3+9^3=737 -> 2*7^3+3^3=713 -> 7^3+1+3^3=371.

Cf. A031179, A046197, A035504, A008585, A165333.

f[n_] := Plus@@(IntegerDigits[n]^3); Trajectory[n_] := Most[NestWhileList[f, n, UnsameQ ,All]]; Select[Range[191], Last[Trajectory[#]]==371 &] (* Ant King, May 24 2013 *)

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

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A008585 a(n) = 3*n.

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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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A035504 Numbers that eventually reach 1 under "x -> sum of cubes of digits of x".

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A165331 Number of iterations until a fixed point or cycle is reached when 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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A154877 Numbers that eventually reach the cycle 160-217-352 under "x -> sum of cubes of digits of x" (see A055012).