A335842 Nonnegative differences of positive cubes and positive tetrahedral numbers.

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

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