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
Keywords
Examples
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.
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Programs
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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)
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).
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