A380749
a(n) is the number of positive integer solutions of n*x*y*z*w = (x + n) * (y + n) * (z + n) * (w + n), x <= y <= z <= w.
Original entry on oeis.org
0, 374, 450, 375, 301, 478, 228, 359, 238, 515, 206, 879, 259, 506, 780, 349, 284, 762, 135, 916, 905, 493, 99, 1189, 423, 306, 318, 869, 70, 1879, 97, 311, 714, 250, 778, 1300, 109, 258, 483, 1334, 71, 1987, 93, 545, 1451, 303, 64, 1156, 202, 504, 481, 822, 71
Offset: 1
For n=5, a(5) = 301 because 5*x*y*z*w = (x + 5)*(y + 5)*(z + 5)*(w + 5), 0 < x <= y <= z <= w has 301 positive integer solutions: {{2,12,596,357595}, {2,12,597,179095}, {2,12,598,119595}, ..., {6,7,9,220}, {6,10,11,20}, {7,9,10,20}}.
-
Table[Length@Solve[n*x*y*z*w == (x + n) (y + n) (z + n) (w + n) &&
0 < x <= y <= z <= w, {x, y, z, w}, Integers], {n, 10}]
A374059
a(n) is the smallest integer k such that k*x*y*z = (x + k) * (y + k) * (z + k), 0 < x <= y <= z has exactly n integer solutions.
Original entry on oeis.org
1, 11, 13, 25, 7, 9, 22, 48, 5, 21, 14, 8, 280, 10, 1020, 4, 70, 3, 6, 240, 2, 42, 12, 660, 30
Offset: 0
For n=8, a(n)=5 because 5 is the smallest integer such that 5*x*y*z = (x+5)*(y+5)*(z+5), 0 < x <= y <= z has exactly 8 positive integer solutions: {{2,12,595}, {2,14,95}, {2,15,70}, {2,20,35}, {3,6,220}, {3,10,20}, {4,5,45}, {5,5,20}}.
A380750
a(n) is the smallest integer k such that k*x*y*z*w = (x + k) * (y + k) * (z + k) * (w + k), 0 < x <= y <= z <= w has exactly n integer solutions.
Original entry on oeis.org
1019, 1559, 1637, 1103, 743, 419, 1039, 359, 311, 479, 653, 509, 389, 251, 593, 521, 263, 197, 1061, 131, 353, 269, 239, 167, 89, 179, 337, 113, 139, 83, 181, 229, 934, 898, 277, 151, 103, 554, 1042, 281, 109, 107, 566, 283, 1299, 79, 386, 157, 1959, 173, 241
Offset: 1
For n=8, a(n)=359 because 359 is the smallest integer such that 359*x*y*z*w = (x+359)*(y+359)*(z+359)*(w+359), 0 < x <= y <= z <= w has exactly 8 positive integer solutions: {{2, 406, 6462, 209302385}, {3, 185, 30515, 357644416}, {4, 168, 1375, 1804333641}, {6, 74, 42001, 462553550}, {6, 97, 1406, 462553550}, {15, 28, 8600, 1804333641}, {15, 100, 168, 1804333641}, {22, 50, 234, 11057989441}}.
A381644
a(n) is the number of positive integer solutions of n*x*y*z*v*w = (x + n) * (y + n) * (z + n) * (v + n) * (w + n), x <= y <= z <= v <= w.
Original entry on oeis.org
0, 21313, 35472, 28901, 36366, 35534, 33048, 55548, 30891, 60741, 76106, 161909, 88494, 114437, 220621, 76856, 56832, 195942, 33510, 212618, 222606, 154046, 21700, 324700, 107022, 94149, 109693, 244884, 35992, 592482, 39051, 134282, 213723, 104829, 363935, 355519, 70334, 110560, 158300, 485946, 46982, 650655
Offset: 1
For n=4, a(4) = 28901 because 4*x*y*z*v*w = (x + 4)*(y + 4)*(z + 4)*(v + 4)*(w + 4), 0 < x <= y <= z <= v <= w has 28901 positive integer solutions: {{2,13,205,42637,1818041676},{2,13,205,42638,909042156},{2,13,205,42639,606042316},{2,13,205,42640,454542396}, ..., {10, 10, 12, 14, 21}, {10, 11, 12, 14, 18}, {10, 12, 12, 14, 16}}.
-
Table[Length@Solve[n*x*y*z*v*w == (x + n) (y + n) (z + n) (v + n) (w + n) &&
0 < x <= y <= z <= v <= w, {x, y, z, v, w}, Integers], {n, 8}]
A382672
Number of integer solutions to Product_{k=1..n} (3 + c(k)) = 3 * Product_{k=1..n} c(k) with 0 < c(k) <= c(k+1).
Original entry on oeis.org
0, 2, 17, 450, 35472, 12127741
Offset: 1
For n=3, a(3) = 17 because 3*x*y*z = (x + 3)*(y + 3)*(z + 3), 0 < x <= y <= z has 17 positive integer solutions: {{2,16,285}, {2,17,150}, {2,18,105}, {2,20,69}, {2,21,60}, {2,24,45}, {2,25,42}, {2,30,33}, {3,7,60}, {3,8,33}, {3,9,24}, {3,12,15}, {4,5,42}, {4,6,21}, {4,7,15}, {5,6,12}, {6,6,9}}.
-
a[n_]:=(p=c/@Range[n];Length@Solve[3 Times@@p==Times@@(3+p)&&LessEqual@@Flatten[{0,p}],p,Integers]);Array[a,5]
A383223
Number of integer solutions to Product_{k=1..n} (4 + c(k)) = 4 * Product_{k=1..n} c(k) with 0 < c(k) <= c(k+1).
Original entry on oeis.org
0, 2, 15, 375, 28901, 5185573
Offset: 1
For n=3, a(3) = 15 because 4*x*y*z = (x + 4)*(y + 4)*(z + 4), 0 < x <= y <= z has 15 positive integer solutions: {{2,13,204}, {2,14,108}, {2,15,76}, {2,16,60}, {2,18,44}, {2,20,36}, {2,24,28}, {3,6,140}, {3,7,44}, {3,8,28}, {3,12,14}, {4,5,36}, {4,6,20}, {4,8,12}, {5,6,12}}.
-
a[n_]:=(p=c/@Range[n]; Length@Solve[4 Times@@p==Times@@(4+p)&&LessEqual@@Flatten[{0, p}], p, Integers]); Array[a, 5]
Showing 1-6 of 6 results.
Comments