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Smallest positive k such that (nk)^3/(n^3+k^3) is an integer, or 0 if no such k exists.
+0
0
0, 2, 6, 4, 0, 3, 0, 8, 18, 10, 0, 6, 0, 14, 30, 16, 0, 9, 0, 20, 42, 22, 0, 12, 0, 26, 54, 28, 0, 15, 0, 32, 66, 34, 0, 18, 0, 38, 78, 40, 0, 14, 0, 44, 90, 46, 0, 24, 0, 50, 102, 52, 0, 27, 0, 56, 114, 58, 0, 30, 0, 62, 126, 64, 260, 33, 0, 68, 138, 70, 0, 36, 0, 74, 150, 76, 0, 39, 0, 80, 162, 82, 0, 28, 0, 86
COMMENTS
For odd n, a(n) is even.
If n = p^j where p is a prime >= 5, then a(n) = 0 (see link for proof). - Robert Israel, Jan 16 2025
PROG
(PARI) A379954(n) = { for(k=1, n^2, if(!(((n*k)^3)%(k^3+n^3)), return(k))); (0); };
Largest k >= 0 such that (nk)^3/(n^3+k^3) is an integer.
+0
0
0, 2, 6, 4, 0, 12, 0, 8, 18, 10, 0, 24, 0, 42, 30, 16, 0, 36, 0, 20, 42, 22, 0, 48, 0, 26, 54, 84, 0, 60, 0, 32, 66, 34, 0, 72, 0, 38, 78, 40, 0, 210, 0, 44, 90, 46, 0, 96, 0, 50, 102, 52, 0, 108, 0, 168, 456, 58, 0, 120, 0, 62, 126, 64, 260, 132, 0, 68, 138, 1330, 0, 144, 0, 74, 150, 76, 0, 1794, 0, 80, 162, 82, 0, 420
COMMENTS
For all n, a(n) < n^2, as for k^3 + n^3 to divide k^3 * n^3, it must also divide n^3 * (k^3 + n^3) - k^3*n^3 = n^6, so k^3 <= n^6 - n^3, and in particular k < n^2. - Robert Israel, Jan 16 2025 Not every a(n) is a multiple of n: a(231) = 616 is the first case where n does not divide a(n).
First odd terms are a(1474) = 6633, a(1628) = 2849 and a(1860) = 5115.
For all odd n, a(n) is even.
If n = p^j where p is a prime >= 5, then a(n) = 0 (see link for proof). - Robert Israel, Jan 16 2025
MAPLE
f:= proc(n) local k;
for k from n^2 by -1 do
if (n*k)^3 mod (n^3 + k^3) = 0 then return k fi
od
end proc:
PROG
(PARI) A379953(n) = forstep(k=n^2, 0, -1, if(!(((n*k)^3)%(k^3+n^3)), return(k)));
0, 1, 0, 1, 1, 0, 1, 0, 1, 0, 1, 1, 0, 1, 0, 1, 0, 1, 0, 1, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 1, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0
COMMENTS
A formula for this sequence would be of great help in understanding the structure of A377091.
In A377091, each block of consecutive terms of the same sign is followed by a jump of magnitude +-s^2 for some integer s>0 to a term of the opposite sign; sequence lists the successive values of s.
+0
0
2, 2, 3, 3, 4, 5, 5, 6, 6, 7, 7, 8, 9, 9, 10, 10, 11, 11, 12, 12, 13, 14, 14, 15, 15, 16, 16, 17, 17, 18, 18, 19, 20, 20, 21, 21, 22, 22, 23, 23, 24, 24, 25, 26, 26, 27, 27, 28, 28, 29, 29, 30, 30, 31, 31, 31, 32, 32, 32, 33, 33, 33, 33, 34, 34, 35, 35, 35, 36, 36, 37, 37, 37, 38, 38, 38, 39, 39, 39, 39, 40, 40, 40, 40, 41, 41
EXAMPLE
In A377091, A377091(12) = 8 is immediately followed by A377091(13) = -8, a negative jump of 16 = 4^2. This the fifth sign change in A377091 (we ignore the initial 0), so a(5) = 4.
Length of n-th block of consecutive negative terms in A377091.
+0
0
2, 2, 8, 6, 6, 8, 8, 10, 10, 15, 23, 14, 16, 16, 18, 18, 20, 20, 22, 22, 24, 50, 26, 28, 28, 29, 31, 5, 26, 32, 2, 33, 34, 4, 32, 37, 37, 4, 34, 3, 35, 2, 40, 42, 42, 44, 89, 2, 46, 45, 2, 46, 2, 50, 50, 55, 102, 2, 53, 56, 56, 58, 58, 60, 60, 124, 64, 64, 66, 66, 68, 68, 70, 70, 144, 74, 74, 76, 76, 78, 78, 83, 159, 82, 168, 86
Length of n-th block of consecutive positive terms in A377091.
+0
0
2, 3, 3, 5, 5, 10, 13, 9, 11, 11, 13, 13, 15, 15, 17, 17, 38, 21, 21, 23, 23, 25, 26, 26, 28, 29, 29, 29, 32, 4, 28, 3, 31, 34, 36, 36, 2, 36, 38, 2, 38, 3, 38, 41, 43, 43, 45, 45, 46, 48, 3, 46, 3, 46, 51, 51, 53, 52, 3, 53, 55, 114, 59, 59, 61, 61, 63, 63, 65, 65, 70, 133, 69, 71, 71, 73, 73, 75, 75, 154, 79, 79, 81, 164, 83, 85
Numbers k > 0 such that both |b(k) - b(k-1)| and |b(k+1) - b(k)| are greater than 1, where b is A377091.
+0
0
21, 32, 43, 52, 74, 91, 112, 133, 147, 161, 162, 184, 211, 212, 242, 243, 273, 308, 343, 381, 422, 463, 464, 508, 509, 553, 554, 602, 651, 652, 653, 678, 704, 758, 759, 760, 761, 813, 814, 815, 816, 820, 821, 822, 828, 872, 873, 931, 932, 938, 966, 998
Terms b(k) (for k > 0, and in order of appearance) such that both |b(k) - b(k-1)| and |b(k+1) - b(k)| are greater than 1, where b is A377091.
+0
0
13, 18, 25, 24, -40, -50, -60, -72, -71, -84, -80, 98, 113, 104, 128, 119, 145, 162, 181, -200, -220, -242, -226, -264, -248, -288, -272, -314, -339, 337, 321, -338, 366, 394, 369, 365, 374, -422, 419, 403, 399, 393, 402, 398, 404, 452, 427, -482, 479, 451, 478, -512
COMMENTS
These are the terms corresponding to the length of runs = 1 in A379880.
8, 11, 14, 18, 22, 26, 29, 32, 38, 40, 41, 45, 49, 50, 55, 56, 61, 64, 67, 76, 80, 84, 85, 90, 91, 96, 97, 116, 127, 128, 129, 139, 150, 172, 173, 174, 175, 197, 198, 199, 200, 202, 203, 204, 207, 222, 223, 246, 247, 250, 262, 264, 276, 289, 290, 301, 302, 304, 307, 313
Sequence Sh of the eight sequences defining the blocks of terms in A377091.
+0
0
1, 3, 12, 18, 24, 32, 39, 49, 59, 71, 98, 112, 128, 144, 162, 180, 199, 219, 241, 263, 286, 339, 363, 390, 422, 448, 482, 483, 508, 545, 543, 578, 612, 613, 645, 685, 723, 724, 760, 758, 796, 799, 839, 881, 923, 966, 1059, 1058, 1104, 1148, 1151, 1196, 1199, 1249, 1299, 1351, 1459, 1457, 1512, 1568, 1624, 1681, 1739, 1799, 1859
COMMENTS
See the comments in A379788 (Sequence Sa) for further information.
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