A004164 -id:A004164 - cp's OEIS frontend

A046459 Dudeney numbers: integers equal to the sum of the digits of their cubes.

0, 1, 8, 17, 18, 26, 27

Offset: 1

Patrick De Geest, Aug 15 1998

This sequence was first found by the French mathematician Claude (Séraphin) Moret-Blanc in 1879. See Le Lionnais page 27 for the last term of this sequence: 27. - Bernard Schott, Dec 07 2012

The name "Dudeney numbers" appears in the October 2018 issue of Mathematics Teacher (see link). - N. J. A. Sloane, Oct 10 2018

			a(3) = 8 because 8^3 = 512 and 5 + 1 + 2 = 8.
a(7) = 27 because 27^3 = 19683 and 1 + 9 + 6 + 8 + 3 = 27.

H. E. Dudeney, 536 Puzzles & Curious Problems, reprinted by Souvenir Press, London, 1968, p. 36, #120.
Italo Ghersi, Matematica dilettevole e curiosa, p. 115, Hoepli, Milano, 1967. [From Vincenzo Librandi, Jan 02 2009]
F. Le Lionnais, Les nombres remarquables, Hermann, 1983.
J. Roberts, Lure of the Integers, The Mathematical Association of America, 1992, p. 172.
David Wells, The Penguin Dictionary of Curious and Interesting Numbers. Penguin Books, NY, 1986, Revised edition 1987. See p. 96.

H. E. Dudeney, 536 Puzzles & Curious Problems.
The Mathematics Teacher, October 2018 Calendar and Solutions, Volume 112, Number 2, October 2018, pages 120 and 122.
ProofWiki, Sequence of Dudeney Numbers.
Bernard Schott and Norbert Verdier, QDL 19: Quels beaux cubes! (French mathematical forum les-mathematiques.net).
Eric Weisstein's World of Mathematics, Cubic Number.

Cf. A004164, A055569, A055575, A055576, A055577.

Cf. A152147.

[n: n in [0..100] | &+Intseq(n^3) eq n ]; // Vincenzo Librandi, Sep 16 2015

Select[Range[0,30],#==Total[IntegerDigits[#^3]]&] (* Harvey P. Dale, Dec 21 2014 *)

isok(k)=sumdigits(k^3)==k \\ Patrick De Geest, Dec 10 2024

a = [n for n in range(100) if sum(map(int, str(n ** 3))) == n] # David Radcliffe, Aug 18 2022

Offset corrected by Arkadiusz Wesolowski, Aug 09 2013

A055565 Sum of digits of n^4.

Original entry on oeis.org

0, 1, 7, 9, 13, 13, 18, 7, 19, 18, 1, 16, 18, 22, 22, 18, 25, 19, 27, 10, 7, 27, 22, 31, 27, 25, 37, 18, 28, 25, 9, 22, 31, 27, 25, 19, 36, 28, 25, 18, 13, 31, 27, 25, 37, 18, 37, 43, 27, 31, 13, 27, 25, 37, 27, 28, 43, 18, 31, 22, 18, 34, 37, 36, 37, 34, 45, 13, 31, 27, 7

Offset: 0

Henry Bottomley, Jun 19 2000

			a(2) = 7 because 2^4 = 16 and 1+6 = 7.

Seiichi Manyama, Table of n, a(n) for n = 0..10000

Cf. A000583, A007953, A055570, A055575 (fixed points), A373914.

Other powers: A004159, A004164, A055566, A055567, A066588.

for i from 0 to 200 do printf(`%d,`,add(j, j=convert(i^4, base, 10))) od;

a[n_Integer]:=Apply[Plus, IntegerDigits[n^4]]; Table[a[n], {n, 0, 100}] (* Vincenzo Librandi, Feb 23 2015 *)

a(n) = sumdigits(n^4); \\ Seiichi Manyama, Nov 16 2021

[sum((n^4).digits()) for n in (0..70)] # Bruno Berselli, Feb 23 2015

a(n) = A007953(A000583(n)). - Michel Marcus, Feb 23 2015

More terms from James Sellers, Jul 04 2000

A055566 Sum of digits of n^5.

Original entry on oeis.org

0, 1, 5, 9, 7, 11, 27, 22, 26, 27, 1, 14, 27, 25, 29, 36, 31, 35, 45, 37, 5, 18, 25, 29, 36, 40, 35, 36, 28, 23, 9, 34, 29, 36, 31, 35, 36, 46, 41, 36, 7, 29, 27, 31, 35, 36, 46, 32, 45, 43, 11, 27, 22, 44, 36, 37, 41, 36, 52, 47, 27, 40, 35, 45, 37, 32, 36, 25, 47, 36, 22, 35

Offset: 0

Henry Bottomley, May 26 2000

			a(2) = 5 because 2^4 = 32 and 3+2 = 5.
Trajectories under the map x->a(x):
1 ->1 ->1 ->1 ->1 ->1 ->1 ->1 ->1 ->..
2 ->5 ->11 ->14 ->29 ->23 ->29 ->23 ->29 ->..
3 ->9 ->27 ->36 ->36 ->36 ->36 ->36 ->36 ->..
4 ->7 ->22 ->25 ->40 ->7 ->22 ->25 ->40 ->..
5 ->11 ->14 ->29 ->23 ->29 ->23 ->29 ->23 ->..
6 ->27 ->36 ->36 ->36 ->36 ->36 ->36 ->36 ->..
7 ->22 ->25 ->40 ->7 ->22 ->25 ->40 ->7 ->..

Seiichi Manyama, Table of n, a(n) for n = 0..10000

Cf. A000012, A004159, A004164, A055565, A055567, A055571, A055576, A066588.

read("transforms") :
A055566 := proc(n)
        digsum(n^5) ;
end proc: # R. J. Mathar, Jul 08 2012

Table[Total[IntegerDigits[n^5]],{n,0,80}] (* Harvey P. Dale, Feb 12 2023 *)

a(n) = sumdigits(n^5); \\ Seiichi Manyama, Nov 16 2021

A070276 Numbers n such that sum of digits of n equals the sum of digits of n^3.

Original entry on oeis.org

0, 1, 8, 10, 80, 100, 171, 378, 468, 487, 577, 585, 586, 684, 800, 1000, 1710, 3780, 4680, 4870, 4877, 5770, 5850, 5851, 5860, 5868, 6840, 8000, 10000, 15877, 17100, 28845, 37800, 46800, 48700, 48770, 57700, 58500, 58510, 58600, 58680, 58968, 59777

Offset: 1

Benoit Cloitre, May 09 2002

Harvey P. Dale, Table of n, a(n) for n = 1..100

Cf. A007953, A004164.

Select[Range[0,60000],Total[IntegerDigits[#]]==Total[IntegerDigits[ #^3]]&] (* Harvey P. Dale, May 10 2012 *)

isok(n) = sumdigits(n) == sumdigits(n^3); \\ Michel Marcus, Aug 12 2017

A055567 Sum of digits of n^6.

Original entry on oeis.org

0, 1, 10, 18, 19, 19, 27, 28, 19, 18, 1, 28, 45, 37, 37, 27, 37, 37, 18, 37, 10, 36, 37, 46, 36, 28, 46, 45, 37, 37, 18, 46, 37, 54, 37, 46, 45, 46, 37, 45, 19, 28, 45, 37, 46, 45, 64, 46, 36, 37, 19, 54, 55, 37, 54, 46, 55, 54, 55, 37, 27, 37, 46, 36, 64, 55, 45, 55, 64, 45

Offset: 0

Henry Bottomley, May 26 2000

			a(2) = 10 because 2^6 = 64 and 6+4 = 10.

Seiichi Manyama, Table of n, a(n) for n = 0..10000

Cf. A000012, A004159, A004164, A055565, A055566, A055572, A055577, A066588.

DigitSum[Range[0, 100]^6] (* Paolo Xausa, Jul 03 2024 *)

a(n) = sumdigits(n^6); \\ Seiichi Manyama, Nov 16 2021

A055569 Sum of digits of a(n)^3 is greater than or equal to a(n).

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 12, 13, 14, 15, 16, 17, 18, 19, 26, 27

Offset: 0

Henry Bottomley, May 26 2000

			a(4) = 4 because 4^3 = 64 and 6+4 = 10>= 4

Cf. A004164, A046459, A055568, A055570, A055571, A055572.

Select[Range[300],Total[IntegerDigits[#^3]]>=#&] (* Harvey P. Dale, Aug 27 2013 *)

A073636 Period 3: repeat [1, 8, 9] ; Digital root of A000578(n) = n^3 for n >= 1.

Original entry on oeis.org

1, 8, 9, 1, 8, 9, 1, 8, 9, 1, 8, 9, 1, 8, 9, 1, 8, 9, 1, 8, 9, 1, 8, 9, 1, 8, 9, 1, 8, 9, 1, 8, 9, 1, 8, 9, 1, 8, 9, 1, 8, 9, 1, 8, 9, 1, 8, 9, 1, 8, 9, 1, 8, 9, 1, 8, 9, 1, 8, 9, 1, 8, 9, 1, 8, 9, 1, 8, 9, 1, 8, 9, 1, 8, 9, 1, 8, 9, 1, 8, 9, 1, 8, 9, 1, 8, 9

Offset: 1

Zak Seidov, Sep 01 2002

a(n) is the decimal expansion of 70/37. [Enrique Pérez Herrero, Jul 28 2009]; corrected by David A. Corneth, Jun 30 2016

Eric Weisstein's World of Mathematics, Digital Root.
Index entries for linear recurrences with constant coefficients, signature (0,0,1).

Cf. A000578, A004164, A010888, A021596. Digital roots of squares are in A056992.

&cat [[1, 8, 9]^^30]; // Wesley Ivan Hurt, Jun 30 2016

seq(op([1, 8, 9]), n=1..50); # Wesley Ivan Hurt, Jun 30 2016

n=3; su[x_] := Sum[IntegerDigits[x][[i]], {i, Length[IntegerDigits[x]]}]; Table[su[su[su[su[x^n]]]], {x, 100}]
NestWhile[Total[IntegerDigits[#]] &, #1, # > 9 &] & /@ (Range[87]^3) (* Jayanta Basu, Jul 03 2013 *)

G.f.: x*(9*x^2+8*x+1)/(1-x^3). - Ant King, Apr 30 2013
From Wesley Ivan Hurt, Jun 30 2016: (Start)
a(n) = a(n-3) for n>3.
a(n) = 6 + 3*cos(2*n*Pi/3) - 7*sin(2*n*Pi/3)/sqrt(3). (End)

Decimal expansion fraction corrected by Ant King, Apr 30 2013

Edited: name specified, offset changed from 0 to 1 (according to name), adjusted formula and g.f. for offset 1, digital root link added. - Wolfdieter Lang, Jan 05 2015

A107679 Numbers n such that sum of digits of n^3 is 2^3 = 8.

Original entry on oeis.org

2, 5, 8, 11, 20, 50, 80, 101, 110, 200, 500, 800, 1001, 1010, 1100, 2000, 5000, 8000, 10001, 10010, 10100, 11000, 20000, 50000, 80000, 100001, 100010, 100100, 101000, 110000, 200000, 500000, 800000, 1000001, 1000010, 1000100, 1001000, 1010000, 1100000

Offset: 1

Zak Seidov, Jun 10 2005

Vincenzo Librandi, Table of n, a(n) for n = 1..50

Cf. A004164 (sum of digits of cubes), A067075, A159462, A159463.

[ n: n in [1..2*10^6] | 8 eq (&+Intseq(n^3)) ]; // Vincenzo Librandi, Aug 13 2017

Do[If[Total[IntegerDigits[m^3]]==8, Print[m]], {m, 2*10^7}] (* Vincenzo Librandi, Aug 13 2017 *)

isok(n) = sumdigits(n^3) == 8; \\ Michel Marcus, Aug 12 2017

More terms from Michel Marcus, Oct 09 2013

A235398 Sum of digits of the cubes of prime numbers.

Original entry on oeis.org

8, 9, 8, 10, 8, 19, 17, 28, 17, 26, 28, 19, 26, 28, 17, 35, 26, 28, 19, 26, 28, 28, 35, 35, 28, 8, 28, 17, 28, 35, 28, 26, 26, 37, 35, 28, 46, 28, 35, 35, 35, 37, 44, 37, 35, 46, 37, 37, 35, 37, 35, 35, 37, 26, 44, 35, 35, 28, 28, 26, 37, 35, 37, 17, 37, 26

Offset: 1

Antonio Graciá Llorente, Jan 09 2014

Cf. A004164, A007953, A123157.

read("transforms") :
A235398 := proc(n)
    digsum(ithprime(n)^3) ;
end proc:
seq(A235398(n),n=1..40) ; # R. J. Mathar, Jul 19 2024

a(n) = sumdigits(prime(n)^3); \\ Michel Marcus, Jan 09 2014

a(n) = A007953(A030078(n)). - R. J. Mathar, Jul 19 2024

A076204 Numbers whose cube has a prime sum of digits.

Original entry on oeis.org

13, 14, 16, 17, 22, 23, 25, 28, 34, 37, 47, 52, 58, 64, 67, 68, 74, 76, 85, 106, 107, 118, 130, 134, 139, 140, 142, 146, 160, 166, 169, 170, 172, 175, 178, 181, 193, 196, 211, 217, 218, 220, 223, 229, 230, 232, 241, 244, 250, 253, 256, 265, 268, 280, 283, 286

Offset: 1

Zak Seidov, Nov 02 2002

Corresponding primes in A109410.

Cf. A004164, A109408.

A076204=Select[Range[500], PrimeQ[Plus@@IntegerDigits[ #^3]]&]

a(n) = (A109408(n))^(1/3).

Edited by N. J. A. Sloane at the suggestion of Andrew S. Plewe, May 21 2007

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A046459 Dudeney numbers: integers equal to the sum of the digits of their cubes.

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A055565 Sum of digits of n^4.

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A055566 Sum of digits of n^5.

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A070276 Numbers n such that sum of digits of n equals the sum of digits of n^3.

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A055567 Sum of digits of n^6.

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A055569 Sum of digits of a(n)^3 is greater than or equal to a(n).

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A073636 Period 3: repeat [1, 8, 9] ; Digital root of A000578(n) = n^3 for n >= 1.

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