A048766 -id:A048766 - cp's OEIS frontend

A000433 n written in base where place values are positive cubes.

0, 1, 2, 3, 4, 5, 6, 7, 10, 11, 12, 13, 14, 15, 16, 17, 20, 21, 22, 23, 24, 25, 26, 27, 30, 31, 32, 100, 101, 102, 103, 104, 105, 106, 107, 110, 111, 112, 113, 114, 115, 116, 117, 120, 121, 122, 123, 124, 125, 126, 127, 130, 131, 132, 200, 201, 202, 203

Offset: 0

R. Muller

Let [d1, d2, d3, ...] be the decimal expansion of the n-th term, then dk is the number of times that the greedy algorithm subtracts the cube k^3 with input n. - Joerg Arndt, Nov 21 2014
For n > 1: A048766(n) = number of digits of a(n); A190311(n) = number of nonzero digits of a(n); A055401(n) = sum of digits of a(n). - Reinhard Zumkeller, May 08 2011
First differs from numbers written in base 8 (A007094) at a(27) = 100, whereas A007094(27) = 33. - Alonso del Arte, Nov 27 2014
The rightmost (least significant) digit never exceeds 7, the second digit from the right never exceeds 3, the third digit never exceeds 2, and the rest are just 0's and 1's. - Ivan Neretin, Sep 03 2015

			a(26) = 32 because 26 = 3 * 2^3 + 2 * 1^3.
a(27) = 100 because 27 = 3^3 + 0 * 2^3 + 0 * 1^3.
a(28) = 101 because 28 = 3^3 + 0 * 2^3 + 1 * 1^3.

Florentin Smarandache, "Properties of the Numbers", University of Craiova Archives, 1975; Arizona State University Special Collections, Tempe, AZ.

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000

Cf. A000578, A007961.

import Data.Char (intToDigit)
a000433 0 = 0
a000433 n = read $ map intToDigit $
  t n $ reverse $ takeWhile (<= n) $ tail a000578_list where
      t _ []          = []
      t m (x:xs)
          | x > m     = 0 : t m xs
          | otherwise = (fromInteger m') : t r xs where (m',r) = divMod m x
-- Reinhard Zumkeller, May 08 2011

A129011 a(n) = floor(n^(4/3)).

Original entry on oeis.org

0, 1, 2, 4, 6, 8, 10, 13, 16, 18, 21, 24, 27, 30, 33, 36, 40, 43, 47, 50, 54, 57, 61, 65, 69, 73, 77, 81, 85, 89, 93, 97, 101, 105, 110, 114, 118, 123, 127, 132, 136, 141, 145, 150, 155, 160, 164, 169, 174, 179, 184, 189, 194, 199, 204, 209, 214, 219, 224, 229, 234

Offset: 0

Jonathan Vos Post, May 01 2007

Churchhouse (1971), as an early example of the use of computers in number theory, conjectured that every positive integer N is the sum of two elements of this sequence and verified the conjecture up to N = 10,000 using the Atlas 1 computer of the Atlas Computer Laboratory at Chilton, U.K. He was able to prove that every sufficiently large integer, N, can be expressed in the form N = floor(n^s) + floor(m^s), n and m being positive integers and s being any number in the interval (1, 4/3). - Peter Bala, Jan 13 2013

J. Spencer, E. Szemeredi and W. T. Trotter, Unit distances in the Euclidean plane, Graph Theory and Combinatorics, B. Bollabas editor, London: Academic Press, 1984, pp. 293-308.

R. Churchhouse, A New Theorem in the Additive Theory of Numbers
P. Erdős, On sets of distances of n points, American Mathematical Monthly 53, pp. 248-250 (1946).
L. Székely, Crossing numbers and hard Erdős problems in discrete geometry, Combin. Probab. Comput. 6(1997).

Cf. A000583, A048766.

Table[ Floor[n^(4/3)], {n, 0, 60}] (* Robert G. Wilson v, May 02 2007 *)

a(n) = floor(n^(4/3)); \\ Altug Alkan, Dec 20 2015

a(n) = sqrtnint(n^4, 3); \\ Michel Marcus, Apr 30 2025

a(n) = floor(n^(4/3)) = A048766(A000583(n)).

More terms from Robert G. Wilson v, May 02 2007

A214081 a(n) = floor( n^(1/3) )!.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24

Offset: 0

Mohammad K. Azarian, Dec 22 2012

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

Cf. A055226, A055228, A214078, A214079, A214080, A214081, A214083.

Cf. A000142, A048766.

PROG(y := [], n := 70, LOOP(IF(n = -1, RETURN y), y := ADJOIN(FLOOR(n^(1/3))!, y), n := n - 1))

[Factorial(Floor(n^(1/3))): n in [0..80]]; // Vincenzo Librandi, Feb 13 2013

Table[Floor[n^(1/3)]!, {n, 0, 100}] (* T. D. Noe, Dec 23 2012 *)
Floor[CubeRoot[Range[0,90]]]! (* Harvey P. Dale, Jan 15 2024 *)

a(n) = floor(n^(1/3))!; \\ Altug Alkan, Jan 11 2016

Sum_{n>=0} 1/a(n) = 10*e. - Amiram Eldar, Aug 15 2022

a(n) = A000142(A048766(n)). - Michel Marcus, Aug 15 2022

A283760 Expansion of (Sum_{i>=1} x^prime(i))*(Sum_{j>=1} x^(j^3)).

Original entry on oeis.org

0, 0, 1, 1, 0, 1, 0, 1, 0, 1, 1, 1, 1, 1, 1, 0, 0, 1, 1, 1, 1, 0, 0, 1, 1, 0, 1, 0, 1, 2, 1, 2, 0, 1, 0, 0, 1, 2, 1, 1, 0, 1, 0, 2, 1, 1, 0, 1, 1, 1, 1, 0, 0, 1, 1, 1, 0, 1, 0, 1, 1, 1, 0, 1, 0, 1, 2, 2, 2, 1, 1, 1, 0, 2, 2, 0, 1, 0, 1, 2, 2, 0, 1, 1, 0, 1, 2, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 2, 0, 1, 1, 1, 0, 1, 2, 1, 1, 1, 1, 2, 2

Offset: 1

Ilya Gutkovskiy, Mar 16 2017

nonn

Number of representations of n as the sum of a prime number and a positive cube.

			a(32) = 2 because 32 = 31 + 1^3 = 5 + 3^3.

Antti Karttunen, Table of n, a(n) for n = 1..10000
Ilya Gutkovskiy, Extended graphical example

Cf. A000040, A000578, A064272, A211167.

nmax = 120; Rest[CoefficientList[Series[Sum[x^Prime[i], {i, 1, nmax}] Sum[x^j^3, {j, 1, nmax}], {x, 0, nmax}], x]]

concat([0,0], Vec((sum(i=1, 120, x^prime(i)) * sum(j=1, 120, x^(j^3))) + O(x^121))) \\ Indranil Ghosh, Mar 16 2017

(define (A283760 n) (cond ((< n 2) 0) (else (let loop ((k (A048766 n)) (s 0)) (if (< k 1) s (loop (- k 1) (+ s (A010051 (- n (expt k 3)))))))))) ;; Antti Karttunen, Aug 18 2017

G.f.: (Sum_{i>=1} x^prime(i))*(Sum_{j>=1} x^(j^3)).

A381042 Alternating sum of floor(n^(1/k)), with k >= 2.

Original entry on oeis.org

0, 0, 0, 0, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 3, 3, 3, 3, 3, 3, 3, 3, 3, 4, 4, 3, 3, 3, 3, 3, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 7, 7, 7, 7, 7

Offset: 0

Friedjof Tellkamp, Apr 14 2025

nonn

			n:       0, 1, 2, 3, 4, 5, 6, 7, 8, 9, ...
k=2 (+): 0, 1, 1, 1, 2, 2, 2, 2, 2, 3, ... (A000196)
k=3 (-): 0, 1, 1, 1, 1, 1, 1, 1, 2, 2, ... (A048766)
...
Sum:     0, 0, 0, 0, 1, 1, 1, 1, 0, 1 ... (= this sequence).

Friedjof Tellkamp, Table of n, a(n) for n = 0..10000

Cf. A000196 (k=2), A048766 (k=3), A255270 (k=4), A178487 (k=5), A178489 (k=6).

Cf. A089361 (nonalternating), A382691, A382692.

z = 100; Table[Sum[(-1)^k Floor[n^(1/k)], {k, 2, 2 Floor@Log[2, z/2] - 1}], {n, 0, z}]

a(n) = A000196(n) - A048766(n) + A255270(n) - A178487(n) + ... .
a(n) = Sum_{k>=2} (-1)^k * floor(n^(1/k)) = Sum_{k>=1} (floor(n^(1/(2*k))) - floor(n^(1/(2*k+1)))).
a(n) = Sum_{i=1..n} A382691(i).
a(n) ~ A382692(n).
G.f.: Sum_{j>=1, k>=2} (-1)^k * x^(j^k)/(1-x).

A064524 Number of noncubes <= n.

Original entry on oeis.org

0, 0, 1, 2, 3, 4, 5, 6, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 60, 61, 62, 63, 64, 65, 66, 67, 68

Offset: 0

Reinhard Zumkeller, Oct 07 2001

nonn

Chai Wah Wu, Table of n, a(n) for n = 0..10000 (terms 0..1000 from Harry J. Smith)

Cf. A007412, A048766.

a(n) = n - sqrtnint(n, 3); \\ Michel Marcus, Jun 18 2024

from sympy import integer_nthroot
def A064524(n): return n-integer_nthroot(n,3)[0] # Chai Wah Wu, Jun 18 2024

a(n) = n - floor(CubicRoot(n)) = n-A048766(n).

G.f.: x/(1 - x)^2 - (1/(1 - x))*Sum_{k>=1} x^(k^3). - Ilya Gutkovskiy, Feb 16 2017

A095396 Modified juggler map: for even numbers: a(n) = floor(n^(2/3)) and for odd n: a(n) = floor(n^(3/2)) = floor(sqrt(n^3)).

Original entry on oeis.org

1, 1, 5, 2, 11, 3, 18, 4, 27, 4, 36, 5, 46, 5, 58, 6, 70, 6, 82, 7, 96, 7, 110, 8, 125, 8, 140, 9, 156, 9, 172, 10, 189, 10, 207, 10, 225, 11, 243, 11, 262, 12, 281, 12, 301, 12, 322, 13, 343, 13, 364, 13, 385, 14, 407, 14, 430, 14, 453, 15, 476, 15, 500, 16, 524, 16, 548, 16

Offset: 1

Labos Elemer, Jun 18 2004

nonn

Parallel to A094683: values for odd arguments are same, for even not necessarily so.

Antti Karttunen, Table of n, a(n) for n = 1..10000

Cf. A000196, A007320, A048766, A094683, A094716, A095397, A095398.

d[x_]:=d[x]=(1-Mod[x, 2])*Floor[N[x^(2/3), 50]] +Mod[x, 2]*Floor[N[x^(3/2), 50]];d[1]=1; Table[d[w], {w, 1, 150}]
Table[If[EvenQ[n],Floor[n^(2/3)],Floor[n^(3/2)]],{n,70}] (* Harvey P. Dale, Dec 28 2018 *)

(define (A095396 n) (if (even? n) (A048766 (* n n)) (A000196 (* n n n)))) ;; Antti Karttunen, May 28 2017

For even n: a(n) = A048766(n^2), for odd n: a(n) = A000196(n^3). - Antti Karttunen, May 28 2017

Name simplified by Antti Karttunen, May 28 2017

A134917 a(n) = ceiling(n^(4/3)).

Original entry on oeis.org

1, 3, 5, 7, 9, 11, 14, 16, 19, 22, 25, 28, 31, 34, 37, 41, 44, 48, 51, 55, 58, 62, 66, 70, 74, 78, 81, 86, 90, 94, 98, 102, 106, 111, 115, 119, 124, 128, 133, 137, 142, 146, 151, 156, 161, 165, 170, 175, 180, 185, 190, 195, 200

Offset: 1

Mohammad K. Azarian, Nov 17 2007

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A048766, A038605, A003059, A134914, A129011, A100196, A121536.

[Ceiling(n^(4/3)): n in [1..60]]; // Vincenzo Librandi, Feb 18 2013

Table[Ceiling[n^(4/3)], {n, 60}] (* Vincenzo Librandi, Feb 18 2013 *)

A190311 Number of nonzero digits when writing n in base where place values are positive cubes, cf. A000433.

Original entry on oeis.org

0, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 2, 1, 2, 2, 1, 2, 2, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 3, 2, 3, 3, 3, 3, 3, 3, 3, 2, 3, 3, 1, 2, 2, 2, 2, 2, 2, 2, 2, 3, 1, 2, 2, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 3, 2, 3, 3, 3, 3, 3

Offset: 0

Reinhard Zumkeller, May 08 2011

For n > 0: a(A000578(n)) = 1; for n > 1: a(A001093(n)) = 2;

a(n) <= A048766(n).

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000

Cf. A190321, A055640.

a190311 n = g n $ reverse $ takeWhile (<= n) $ tail a000578_list where
  g _ []                 = 0
  g m (x:xs) | x > m     = g m xs
             | otherwise = signum m' + g r xs where (m',r) = divMod m x

A276614 The infinite trunk of greedy cubes beanstalk with reversed subsections.

Original entry on oeis.org

0, 7, 26, 21, 14, 63, 59, 52, 47, 40, 33, 124, 115, 110, 103, 96, 89, 84, 77, 70, 215, 208, 201, 194, 187, 183, 176, 171, 164, 157, 150, 145, 138, 131, 342, 339, 330, 318, 311, 304, 299, 292, 285, 278, 274, 267, 262, 255, 248, 241, 236, 229, 222, 511, 506, 499, 492, 487, 480, 473, 466, 457, 445, 438, 431, 426, 419, 412, 405, 401

Offset: 0

Antti Karttunen, Sep 07 2016 and Sep 09 2016

nonn

Antti Karttunen, Table of n, a(n) for n = 0..10236

Cf. A000578, A010057, A048766, A261225, A276612, A276613.

(definec (A276614 n) (cond ((zero? n) n) ((= n 1) 7) (else (let ((maybe_next (A261225 (A276614 (- n 1))))) (if (zero? (A010057 (+ 1 maybe_next))) maybe_next (+ -1 (A000578 (+ 2 (A048766 (+ 1 maybe_next))))))))))

a(0) = 0; a(1) = 7; for n > 1, if A261225(a(n-1))+1 is not a cube, then a(n) = A261225(a(n-1)), otherwise a(n) = A000578(2+A048766(A261225(a(n-1)))) - 1.

cp's OEIS Frontend

A000433 n written in base where place values are positive cubes.

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A129011 a(n) = floor(n^(4/3)).

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A214081 a(n) = floor( n^(1/3) )!.

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A283760 Expansion of (Sum_{i>=1} x^prime(i))*(Sum_{j>=1} x^(j^3)).

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A381042 Alternating sum of floor(n^(1/k)), with k >= 2.

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A064524 Number of noncubes <= n.

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A095396 Modified juggler map: for even numbers: a(n) = floor(n^(2/3)) and for odd n: a(n) = floor(n^(3/2)) = floor(sqrt(n^3)).

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