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
COMMENTS
A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798. The median of a multiset is either the middle part (for odd length), or the average of the two middle parts (for even length).
EXAMPLE
The 0-prepended prime indices of 1617 are {0,2,4,4,5}, with sorted differences {0,1,2,2}, with median 3/2, so 1617 is not in the sequence.
MATHEMATICA
prix[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];
Select[Range[2, 100], IntegerQ[Median[Differences[Prepend[prix[#], 0]]]]&]
CROSSREFS
For mean instead of median we have A340610. Positions of even terms in A360555. The complement is A360557 (without 1). These partitions are counted by A360688. A360614/ A360615 = mean of first differences of 0-prepended prime indices.
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
COMMENTS
None of these partitions is strict.
Also the number of integer partitions of n which, after appending 0, have first differences of median 0.
EXAMPLE
The a(3) = 1 through a(9) = 10 partitions:
(111) (1111) (11111) (222) (22111) (2222) (333)
(21111) (31111) (22211) (22221)
(111111) (211111) (41111) (33111)
(1111111) (221111) (51111)
(311111) (222111)
(2111111) (411111)
(11111111) (2211111)
(3111111)
(21111111)
(111111111)
For example, the partition y = (4,4,3,1,1,1,1) has 0-appended differences (0,1,2,0,0,0,0), with median 0, so y is counted under a(15).
MATHEMATICA
Table[Length[Select[IntegerPartitions[n], Length[#]>2*Length[Union[#]]&]], {n, 0, 30}]
CROSSREFS
The non-prepended version is A237363. These partitions have ranks A360558. For any integer median (not just 0) we have A360688. A008284 counts partitions by number of parts.
A116608 counts partitions by number of distinct parts.
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
COMMENTS
A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798. The median of a multiset is either the middle part (for odd length), or the average of the two middle parts (for even length).
EXAMPLE
The 0-prepended prime indices of 1617 are {0,2,4,4,5}, with sorted differences {0,1,2,2}, with median 3/2, so 1617 is in the sequence.
MATHEMATICA
prix[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];
Select[Range[2, 100], !IntegerQ[Median[Differences[Prepend[prix[#], 0]]]]&]
CROSSREFS
For mean instead of median complement we have A340610, counted by A168659. These partitions are counted by A360691. A287352 lists 0-prepended first differences of prime indices.
A355536 lists first differences of prime indices.
A360614/ A360615 = mean of first differences of 0-prepended prime indices.
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
COMMENTS
A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.
EXAMPLE
The 0-prepended prime indices of 100 are {0,1,1,3,3}, with differences (1,0,2,0), with mean 3/4, so a(100) = 3.
MATHEMATICA
prix[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];
Table[If[n==1, 0, Numerator[Mean[Differences[Prepend[prix[n], 0]]]]], {n, 100}]
PROG
(PARI) A360614(n) = if(1==n, 0, my(u=primepi(vecmax(factor(n)[, 1]))); (u/gcd(u, bigomega(n)))); \\ Antti Karttunen, Oct 23 2023
CROSSREFS
For twice median instead of mean we have A360555. A316413 lists numbers with integer mean prime index, complement A348551.
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
COMMENTS
The median of a multiset is either the middle part (for odd length), or the average of the two middle parts (for even length).
EXAMPLE
The a(1) = 1 through a(8) = 16 partitions:
(1) (2) (3) (4) (5) (6) (7) (8)
(11) (21) (22) (32) (33) (43) (44)
(111) (31) (41) (42) (52) (53)
(1111) (2111) (51) (61) (62)
(11111) (222) (421) (71)
(321) (2221) (431)
(2211) (3211) (521)
(3111) (4111) (2222)
(111111) (211111) (3221)
(1111111) (3311)
(4211)
(5111)
(32111)
(221111)
(311111)
(11111111)
For example, the partition y = (3,2,2,1) has multiplicities (1,2,1), and the multiset {1,1,2} has median 1, so y is counted under a(8).
MATHEMATICA
Table[Length[Select[IntegerPartitions[n], IntegerQ[Median[Length/@Split[#]]]&]], {n, 30}]
CROSSREFS
The case of an odd number of multiplicities is A090794. These partitions have ranks A360553.
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
COMMENTS
The median of a multiset is either the middle part (for odd length), or the average of the two middle parts (for even length).
EXAMPLE
The a(1) = 1 through a(8) = 16 partitions:
(1) (2) (3) (4) (5) (6) (7) (8)
(11) (111) (22) (311) (33) (331) (44)
(31) (11111) (42) (421) (53)
(1111) (51) (511) (62)
(222) (3211) (71)
(321) (31111) (422)
(3111) (1111111) (431)
(111111) (521)
(2222)
(3221)
(3311)
(4211)
(5111)
(32111)
(311111)
(11111111)
For example, the partition y = (7,4,2,1,1) has distinct parts {1,2,4,7} with median 3, so y is counted under a(15).
MATHEMATICA
Table[Length[Select[IntegerPartitions[n], IntegerQ[Median[Union[#]]]&]], {n, 30}]
CROSSREFS
For multiplicities instead of distinct parts: A360687. The complement is counted by A360689. A000975 counts subsets with integer median.
A116608 counts partitions by number of distinct parts.
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
COMMENTS
A number's (unordered) prime signature (row n of A118914) is the multiset of positive exponents in its prime factorization. The median of a multiset is either the middle part (for odd length), or the average of the two middle parts (for even length).
EXAMPLE
The terms together with their prime indices begin:
1: {}
2: {1}
6: {1,2}
30: {1,2,3}
42: {1,2,4}
49: {4,4}
60: {1,1,2,3}
66: {1,2,5}
70: {1,3,4}
78: {1,2,6}
84: {1,1,2,4}
90: {1,2,2,3}
For example, the prime indices of 2760 are {1,1,1,2,3,9}. The signature is (3,1,1,1), with median 1. The first differences of 0-prepended prime indices are (1,0,0,1,1,6), with median 1/2. So 2760 is not in the sequence.
MATHEMATICA
prix[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];
Select[Range[100], Median[Length/@Split[prix[#]]] == Median[Differences[Prepend[prix[#], 0]]]&]
CROSSREFS
For distinct prime indices instead of 0-prepended differences: A360453. For mean instead of median we have A360680. Multisets with integer median:
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
COMMENTS
The median of a multiset is either the middle part (for odd length), or the average of the two middle parts (for even length).
EXAMPLE
The a(3) = 1 through a(9) = 13 partitions:
(111) (1111) (11111) (222) (22111) (2222) (333)
(321) (31111) (3221) (432)
(2211) (211111) (3311) (531)
(21111) (1111111) (22211) (22221)
(111111) (32111) (33111)
(41111) (51111)
(221111) (222111)
(311111) (321111)
(2111111) (411111)
(11111111) (2211111)
(3111111)
(21111111)
(111111111)
MATHEMATICA
Table[Length[Select[IntegerPartitions[n], Median[Differences[#, 2]]==0&]], {n, 0, 30}]
CROSSREFS
For first differences we have A237363. For sum instead of median we have A360683. A008284 counts partitions by number of parts.
A360005 gives median of prime indices (times two).
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
COMMENTS
The median of a multiset is either the middle part (for odd length), or the average of the two middle parts (for even length).
EXAMPLE
The a(1) = 0 through a(9) = 8 partitions:
. . . (211) (221) (411) (322) (332) (441)
(311) (21111) (331) (422) (522)
(511) (611) (711)
(22111) (22211) (22221)
(31111) (41111) (33111)
(2111111) (51111)
(2211111)
(3111111)
For example, the partition y = (3,2,2,1) has multiplicities (1,2,1), and the multiset {1,1,2} has median 1, so y is not counted under a(8).
MATHEMATICA
Table[Length[Select[IntegerPartitions[n], !IntegerQ[Median[Length/@Split[#]]]&]], {n, 30}]
CROSSREFS
These partitions have ranks A360554. A360069 = partitions with integer mean of multiplicities, ranks A067340.
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
COMMENTS
All of these partitions have even length.
The median of a multiset is either the middle part (for odd length), or the average of the two middle parts (for even length).
EXAMPLE
The a(1) = 0 through a(10) = 10 partitions:
. (11) . (31) (32) (33) (52) (53) (54) (55)
(2111) (51) (2221) (71) (72) (73)
(2211) (4111) (3311) (3222) (91)
(3111) (5111) (6111) (3322)
(321111) (3331)
(4411)
(5311)
(7111)
(322111)
(421111)
MATHEMATICA
Table[Length[Select[IntegerPartitions[n], !IntegerQ[Median[Differences[Prepend[Reverse[#], 0]]]]&]], {n, 30}]
CROSSREFS
The complement is counted by A360688. A008284 counts partitions by number of parts.
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