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A056733 Each number is the sum of the cubes of its 3 sections. 4
153, 370, 371, 407, 165033, 221859, 336700, 336701, 340067, 341067, 407000, 407001, 444664, 487215, 982827, 983221, 166500333, 296584415, 333667000, 333667001, 334000667, 710656413, 828538472, 142051701000, 166650003333, 262662141664, 333366670000 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

The first four terms are also called Narcissistic or Armstrong numbers. The first 16 terms are found in the Spencer's book, pages 65 and 101.

The sequence contains several infinite subsequences such as 153, 165033, 166500333, 166650003333, ...; 370, 336700, 333667000, 333366670000, ... or 371, 336701, 333667001, 333366670001, ... - Ulrich Schimke (ulrschimke(AT)aol.com), Jun 08 2001

From Daniel Forgues, Jan 30 2015: (Start)

The subsequence {153, 165033, 166500333, ...} are numbers of the form

  [(10^n - 4) / 6] * (10^n)^2 + [(10^n) / 2] * (10^n)^1 +

  [(10^n - 1) / 3] * (10^n)^0 =

  [(10^n - 4) / 6]^3 + [(10^n) / 2]^3 + [(10^n - 1) / 3]^3, n >= 1,

thus equal to the sum of the cube of their "digits" in base 10^n.

The subsequence {370, 336700, 333667000, ...} are numbers of the form

  [(10^n - 1) / 3] * (10^n)^2 + {10^n - [(10^n - 1) / 3]} * (10^n)^1 =

  [(10^n - 1) / 3]^3 + {10^n - [(10^n - 1) / 3]}^3, n >= 1,

thus equal to the sum of their "digits" in base 10^n.

The subsequence {371, 336701, 333667001, ...} is trivially derived from the subsequence {370, 336700, 333667000, ...}, since 1^3 = 1.

The subsequence {407, 340067, 334000667, ... } are numbers of the form

  {10^n - 2 * [(10^n - 1) / 3]} * (10^n)^2 +

  {10^n - [(10^n - 1) / 3]} * (10^n)^0 =

  {10^n - 2 * [(10^n - 1) / 3]}^3 + {10^n - [(10^n - 1) / 3]}^3, n >= 1,

thus equal to the sum of their "digits" in base 10^n.

"There are just four numbers (after 1) which are the sums of the cubes of their digits, viz. 153 = 1^3 + 5^3 + 3^3, 370 = 3^3 + 7^3 + 0^3, 371 = 3^3 + 7^3 + 1^3, and 407 = 4^3 + 0^3 + 7^3. This is an odd fact, very suitable for puzzle columns and likely to amuse amateurs, but there is nothing in it which appeals much to a mathematician. The proof is neither difficult nor interesting--merely a little tiresome. The theorem is not serious; and it is plain that one reason (though perhaps not the most important) is the extreme speciality of both the enunciation and the proof, which is not capable of any significant generalization." -- G. H. Hardy, "A Mathematician’s Apology" (End)

From Daniel Forgues, Feb 04 2015: (Start)

The subsequence {341067, 333401006667, 333334001000666667, ...} is trivially derived from the even indexed terms 2n, n >= 1, of the subsequence {407, 340067, 334000667, 333400006667, ...}, since (10^n)^3 = 10^n * 10^(2n). These numbers are equal to the sum of the cube of their "digits" in base 10^(2n), n >= 1.

The number 407000 is trivially derived from 407, since 40^3 + 70^3 =

(4 * 10)^3 + (7 * 10)^3 = (4^3 + 7^3) * 10^3 = 407 * 1000 = 407000.

The number 407001 is trivially derived from 407000, since 1^3 = 1. (End)

From Jose M. Arenas, Mar 08 2017: (Start)

The subsequence {340067000000, 334000667000000000, 333400006667000000000000, ...} are numbers of the form

  (4 * 10^(n + 2) + ((10^(n + 1) - 1) / 3) * 10^(n + 3)) * 10^(4 * n + 8) +

  (7 * 10^(n + 2) + (2 * (10^(n + 1) - 1) / 3) * 10^(n + 3)) * 10^(2 * n + 4) =

  (4 * 10^(n + 2) + ((10^(n + 1) - 1) / 3) * 10^(n + 3))^3 +

  (7 * 10^(n + 2) + (2 * (10^(n + 1) - 1) / 3) * 10^(n + 3))^3, n >= 0,

thus equal to the sum of their 3 sections, each section of (2 * n + 4) digits.

The subsequence {340067000001, 334000667000000001, 333400006667000000000001, ...} is trivially derived from the subsequence {340067000000, 334000667000000000, 333400006667000000000000, ...}, since 1^3 = 1.

(End)

REFERENCES

J. S. Madachy, Madachy's Mathematical Recreations, pp. 166 Dover NY 1979.

Donald D. Spencer, "Exploring number theory with microcomputers", pp. 65 and 101, Camelot Publishing Co.

LINKS

Jose M. Arenas and Giovanni Resta, Table of n, a(n) for n = 1..69 (terms < 10^18, first 49 terms from Jose M. Arenas)

Jose M. Arenas, Python code

EXAMPLE

333667001 = 333^3 + 667^3 + 001^3.

MATHEMATICA

f[n_] := Block[{len = IntegerLength@ n}, If[IntegerQ[len/3], n == Plus @@ Flatten[(FromDigits /@ Partition[IntegerDigits@ n, len/3])^3], False]]; Select[Range[10^6], f] (* Michael De Vlieger, Jan 31 2015 *)

PROG

(Python)

def a():

..n = 1

..while n < 10**9:

....st = str(n)

....if len(st) % 3 == 0:

......s1 = st[:int(len(st)/3)]

......s2 = st[int(len(st)/3):int(2*len(st)/3)]

......s3 = st[int(2*len(st)/3):int(len(st))]

......if int(s1)**3+int(s2)**3+int(s3)**3 == int(st):

........print(n, end=', ')

........n += 1

......else:

........n += 1

....else:

......n = 10*n

a()

# Derek Orr, Jul 03 2014

(Python)

def a():

..for i in range(1, 10):

....for j in range(10):

......for k in range(10):

........if i**3 + j**3 + k**3 == i*100 + j*10 + k:

..........print(i*100 + j*10 + k)

..for i in range(10, 100):

....for j in range(100):

......for k in range(100):

........if i**3 + j**3 + k**3 == i*10000 + j*100 + k:

..........print(i*10000 + j*100 + k)

..for i in range(100, 1000):

....for j in range(1000):

......for k in range(1000):

........if i**3 + j**3 + k**3 == i*1000000 + j*1000 + k:

..........print(i*1000000 + j*1000 + k)

a()

# Denys Contant, Feb 23 2017

CROSSREFS

Cf. A005188.

See A271730 for a related sequence.

Sequence in context: A066528 A046197 A271730 * A256748 A256741 A246629

Adjacent sequences:  A056730 A056731 A056732 * A056734 A056735 A056736

KEYWORD

nonn,base

AUTHOR

Carlos Rivera, Aug 13 2000

EXTENSIONS

Offset changed to 1 by N. J. A. Sloane, Jul 07 2014

a(24)-a(27) from Jose M. Arenas, Mar 08 2017

STATUS

approved

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Last modified October 19 16:17 EDT 2019. Contains 328223 sequences. (Running on oeis4.)