login
The OEIS Foundation is supported by donations from users of the OEIS and by a grant from the Simons Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A335842 Nonnegative differences of positive cubes and positive tetrahedral numbers. 1
0, 1, 4, 5, 7, 8, 17, 23, 26, 29, 31, 36, 41, 44, 49, 51, 54, 57, 60, 63, 68, 69, 77, 83, 90, 93, 96, 99, 105, 115, 121, 122, 123, 124, 132, 144, 148, 149, 151, 160, 169, 173, 178, 180, 181, 184, 188, 191, 196, 206, 211, 212, 215, 223, 226, 230, 258, 259, 274 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,3

COMMENTS

The sequence is the difference between the cubic number (A000578) and the tetrahedral number (A000292) such that terms are of the form A000578(i) - A000292(j), where A000578(i) >= A000292(j) >= 0.

It appears that, for a(n) > 456, the number of terms up to a(n) in this sequence is smaller than the number of prime numbers less than or equal to a(n), or n < pi(a(n)), where pi is the prime counting function. See the figure attached in the Links section.

LINKS

Table of n, a(n) for n=1..59.

Ya-Ping Lu, The number of terms up to a(n) and the number of prime numbers less than or equal to a(n)

FORMULA

The difference between the i-th cubic number, c(i), and j-th tetrahedral number, t(j), is d = i^3 - j*(j+1)*(j+2)/6, where i, j >=1 and c(i) >= t(j).

EXAMPLE

a(1)=0 because c(1)-t(1) = 1-1 = 0;

a(2)=1 because c(11)-t(19) = 1331-1330 = 1;

a(5)=7 because c(2)-t(1) = 8-1 = 7, and c(3)-t(4) = 27-20 = 7;

a(18)=57 because c(7)-t(11) = 343-286 = 57, and c(8)-t(13) = 512-455 = 57;

a(26)=93 because c(2313)-t(4202) = 12374478297-12374478204 = 93.

PROG

(Python)

import math

n_max = 10000000

d_max = 10000

list1 = []

n = 1

while n <= n_max:

  a_tetr = n*(n + 1)*(n + 2)//6

  m_min = math.floor(math.pow(a_tetr, 1/3))

  m = m_min

  a_cube_max = n*(n + 1)*(n + 2)//6 + d_max

  m_max = math.ceil(math.pow(a_cube_max, 1/3))

  while m <= m_max:

      a_cube = m**3

      d = a_cube - a_tetr

      if d >= 0 and d <= d_max and d not in list1:

          list1.append(d)

      m += 1

  n += 1

list1.sort()

print(list1)

CROSSREFS

Cf. A000292, A000578, A175034, A175035, A335761.

Sequence in context: A033164 A276324 A226628 * A140076 A135186 A011336

Adjacent sequences:  A335839 A335840 A335841 * A335843 A335844 A335845

KEYWORD

nonn

AUTHOR

Ya-Ping Lu, Jun 26 2020

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified May 15 12:37 EDT 2021. Contains 343920 sequences. (Running on oeis4.)