A335842 - OEIS

%I #20 Aug 17 2020 23:17:47

%S 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,

%T 93,96,99,105,115,121,122,123,124,132,144,148,149,151,160,169,173,178,

%U 180,181,184,188,191,196,206,211,212,215,223,226,230,258,259,274

%N Nonnegative differences of positive cubes and positive tetrahedral numbers.

%C 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.

%C 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.

%H Ya-Ping Lu, <a href="/A335842/a335842.pdf">The number of terms up to a(n) and the number of prime numbers less than or equal to a(n)</a>

%F 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).

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

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

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

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

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

%o (Python)

%o import math

%o n_max = 10000000

%o d_max = 10000

%o list1 = []

%o n = 1

%o while n <= n_max:

%o   a_tetr = n*(n + 1)*(n + 2)//6

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

%o   m = m_min

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

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

%o   while m <= m_max:

%o       a_cube = m**3

%o       d = a_cube - a_tetr

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

%o           list1.append(d)

%o       m += 1

%o   n += 1

%o list1.sort()

%o print(list1)

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

%K nonn

%O 1,3

%A _Ya-Ping Lu_, Jun 26 2020