Displaying 1-10 of 11 results found.
Numbers > 1 whose first differences of 0-prepended prime indices have integer median.
+10
20
2, 3, 5, 6, 7, 8, 9, 11, 12, 13, 14, 16, 17, 18, 19, 20, 21, 23, 26, 27, 28, 29, 30, 31, 32, 35, 37, 38, 39, 41, 42, 43, 44, 45, 47, 48, 49, 50, 52, 53, 57, 58, 59, 60, 61, 63, 64, 65, 66, 67, 68, 70, 71, 72, 73, 74, 75, 76, 78, 79, 80, 81, 83, 84, 86, 87, 89
CROSSREFS
These partitions are counted by A360688.
Number of integer partitions of n with more adjacent equal parts than distinct parts.
+10
17
0, 0, 0, 1, 1, 1, 3, 4, 7, 10, 12, 18, 28, 36, 52, 68, 92, 119, 161, 204, 269, 355, 452, 571, 738, 921, 1167, 1457, 1829, 2270, 2834, 3483, 4314, 5300, 6502, 7932, 9665, 11735, 14263, 17227, 20807, 25042, 30137, 36099, 43264, 51646, 61608, 73291, 87146, 103296
CROSSREFS
For any integer median (not just 0) we have A360688.
Numbers > 1 whose sorted first differences of 0-prepended prime indices have non-integer median.
+10
13
4, 10, 15, 22, 24, 25, 33, 34, 36, 40, 46, 51, 54, 55, 56, 62, 69, 77, 82, 85, 88, 93, 94, 100, 104, 115, 118, 119, 121, 123, 134, 135, 136, 141, 146, 152, 155, 161, 166, 177, 184, 187, 194, 196, 201, 205, 206, 217, 218, 219, 220, 221, 225, 232, 235, 240, 248
Numerator of the average distance between consecutive 0-prepended prime indices of n; a(1) = 0.
+10
11
0, 1, 2, 1, 3, 1, 4, 1, 1, 3, 5, 2, 6, 2, 3, 1, 7, 2, 8, 1, 2, 5, 9, 1, 3, 3, 2, 4, 10, 1, 11, 1, 5, 7, 2, 1, 12, 4, 3, 3, 13, 4, 14, 5, 1, 9, 15, 2, 2, 1, 7, 2, 16, 1, 5, 1, 4, 5, 17, 3, 18, 11, 4, 1, 3, 5, 19, 7, 9, 4, 20, 2, 21, 6, 1, 8, 5, 2, 22, 3, 1, 13, 23, 1, 7, 7, 5, 5, 24, 3, 3, 3, 11, 15, 4, 1, 25, 4, 5, 3
Number of integer partitions of n whose multiplicities have integer median.
+10
11
1, 2, 3, 4, 5, 9, 10, 16, 22, 34, 42, 65, 80, 115, 145, 195, 240, 324, 396, 519, 635, 814, 994, 1270, 1549, 1952, 2378, 2997, 3623, 4521, 5466, 6764, 8139, 10008, 12023, 14673, 17534, 21273, 25336, 30593, 36302, 43575, 51555, 61570, 72653, 86382, 101676
Number of integer partitions of n whose distinct parts have integer median.
+10
9
1, 2, 2, 4, 3, 8, 7, 16, 17, 31, 35, 60, 67, 99, 121, 170, 200, 270, 328, 436, 522, 674, 828, 1061, 1292, 1626, 1983, 2507, 3035, 3772, 4582, 5661, 6801, 8358, 10059, 12231, 14627, 17702, 21069, 25423, 30147, 36100, 42725, 50936, 60081, 71388, 84007, 99408
Numbers for which the prime signature has the same median as the first differences of 0-prepended prime indices.
+10
4
1, 2, 6, 30, 42, 49, 60, 66, 70, 78, 84, 90, 102, 105, 114, 120, 126, 132, 138, 140, 150, 154, 156, 168, 174, 186, 198, 204, 210, 222, 228, 234, 246, 258, 264, 270, 276, 280, 282, 286, 294, 306, 308, 312, 315, 318, 330, 342, 348, 350, 354, 366, 372, 378, 385
Number of integer partitions of n of length > 2 whose second differences have median 0.
+10
3
0, 0, 0, 1, 1, 1, 5, 4, 10, 13, 18, 23, 44, 44, 72, 98, 132, 162, 241, 277, 394, 497, 643, 800, 1076, 1287, 1660, 2078, 2604, 3192, 4065, 4892, 6113, 7490, 9166, 11110, 13717, 16429, 20033, 24201, 29143, 34945, 42251, 50219, 60253, 71852, 85503, 101501, 120899
Number of integer partitions of n with non-integer median of multiplicities.
+10
2
0, 0, 0, 1, 2, 2, 5, 6, 8, 8, 14, 12, 21, 20, 31, 36, 57, 61, 94, 108, 157, 188, 261, 305, 409, 484, 632, 721, 942, 1083, 1376, 1585, 2004, 2302, 2860, 3304, 4103, 4742, 5849, 6745, 8281, 9599, 11706, 13605, 16481, 19176, 23078, 26838, 32145, 37387, 44465
Number of integer partitions of n with non-integer median of 0-prepended first differences.
+10
2
0, 1, 0, 1, 2, 4, 3, 4, 5, 10, 10, 15, 22, 26, 34, 42, 57, 63, 85, 105, 121, 149, 202, 230, 305, 355, 459, 544, 687, 778, 991, 1130, 1396, 1598, 1947, 2258, 2761, 3143, 3820, 4412, 5330, 6104, 7404, 8499, 10105, 11694, 13922, 15917, 18904, 21646, 25462, 29213
CROSSREFS
The complement is counted by A360688.
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