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 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 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

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