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One and numbers whose multiset of prime factors (with multiplicity) has the same median as maximum.
+10
10
1, 2, 3, 4, 5, 7, 8, 9, 11, 13, 16, 17, 18, 19, 23, 25, 27, 29, 31, 32, 37, 41, 43, 47, 49, 50, 53, 54, 59, 61, 64, 67, 71, 73, 75, 79, 81, 83, 89, 97, 98, 101, 103, 107, 108, 109, 113, 121, 125, 127, 128, 131, 137, 139, 147, 149, 151, 157, 162, 163, 167, 169
COMMENTS
First differs from A334965 in having 750 and lacking 2250. 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 prime factorization of 108 is 2*2*3*3*3, and the multiset {2,2,3,3,3} has median 3 and maximum 3, so 108 is in the sequence.
The prime factorization of 2250 is 2*3*3*5*5*5, and the multiset {2,3,3,5,5,5} has median 4 and maximum 5, so 2250 is not in the sequence.
The terms together with their prime indices begin:
1: {} 25: {3,3} 64: {1,1,1,1,1,1}
2: {1} 27: {2,2,2} 67: {19}
3: {2} 29: {10} 71: {20}
4: {1,1} 31: {11} 73: {21}
5: {3} 32: {1,1,1,1,1} 75: {2,3,3}
7: {4} 37: {12} 79: {22}
8: {1,1,1} 41: {13} 81: {2,2,2,2}
9: {2,2} 43: {14} 83: {23}
11: {5} 47: {15} 89: {24}
13: {6} 49: {4,4} 97: {25}
16: {1,1,1,1} 50: {1,3,3} 98: {1,4,4}
17: {7} 53: {16} 101: {26}
18: {1,2,2} 54: {1,2,2,2} 103: {27}
19: {8} 59: {17} 107: {28}
23: {9} 61: {18} 108: {1,1,2,2,2}
MATHEMATICA
Select[Range[100], (y=Flatten[Apply[ConstantArray, FactorInteger[#], {1}]]; Max@@y==Median[y])&]
CROSSREFS
Partitions of this type are counted by A053263. For parts at middle position (instead of median) we have A362622. A362611 counts modes in prime factorization, triangle version A362614.
A362613 counts co-modes in prime factorization, triangle version A362615.
Number of integer partitions of n whose run-lengths are either strictly increasing or strictly decreasing.
+10
7
1, 1, 2, 2, 4, 5, 7, 10, 13, 15, 21, 26, 29, 39, 49, 50, 68, 80, 92, 109, 129, 142, 181, 201, 227, 262, 317, 343, 404, 456, 516, 589, 677, 742, 870, 949, 1077, 1207, 1385, 1510, 1704, 1895, 2123, 2352, 2649, 2877, 3261, 3571, 3966, 4363, 4873, 5300, 5914, 6466
EXAMPLE
The a(1) = 1 through a(8) = 13 partitions:
(1) (2) (3) (4) (5) (6) (7) (8)
(11) (111) (22) (221) (33) (322) (44)
(211) (311) (222) (331) (332)
(1111) (2111) (411) (511) (422)
(11111) (3111) (2221) (611)
(21111) (4111) (2222)
(111111) (22111) (5111)
(31111) (22211)
(211111) (41111)
(1111111) (221111)
(311111)
(2111111)
(11111111)
MATHEMATICA
Table[Length[Select[IntegerPartitions[n], Or[Less@@Length/@Split[#], Greater@@Length/@Split[#]]&]], {n, 0, 30}]
CROSSREFS
The generalization to compositions is A333191. Partitions with distinct run-lengths are A098859. Partitions with strictly increasing run-lengths are A100471. Partitions with strictly decreasing run-lengths are A100881. Partitions with weakly decreasing run-lengths are A100882. Partitions with weakly increasing run-lengths are A100883. Partitions with unimodal run-lengths are A332280. Partitions whose run-lengths are not increasing nor decreasing are A332641. Compositions whose run-lengths are unimodal or co-unimodal are A332746. Compositions that are neither increasing nor decreasing are A332834. Strictly increasing or strictly decreasing compositions are A333147. Compositions with strictly increasing run-lengths are A333192. Numbers with strictly increasing prime multiplicities are A334965.
Number of compositions of n with strictly increasing run-lengths.
+10
5
1, 1, 2, 2, 4, 5, 7, 10, 14, 16, 24, 31, 37, 51, 67, 76, 103, 129, 158, 199, 242, 293, 370, 450, 538, 652, 799, 953, 1147, 1376, 1635, 1956, 2322, 2757, 3271, 3845, 4539, 5336, 6282, 7366, 8589, 10046, 11735, 13647, 15858, 18442, 21354, 24716, 28630, 32985
COMMENTS
A composition of n is a finite sequence of positive integers summing to n.
EXAMPLE
The a(1) = 1 through a(8) = 14 compositions:
(1) (2) (3) (4) (5) (6) (7) (8)
(11) (111) (22) (122) (33) (133) (44)
(211) (311) (222) (322) (233)
(1111) (2111) (411) (511) (422)
(11111) (3111) (1222) (611)
(21111) (4111) (2222)
(111111) (22111) (5111)
(31111) (11222)
(211111) (41111)
(1111111) (122111)
(221111)
(311111)
(2111111)
(11111111)
For example, the composition (1,2,2,1,1,1) has run-lengths (1,2,3), so is counted under a(8).
MATHEMATICA
Table[Length[Select[Join@@Permutations/@IntegerPartitions[n], Less@@Length/@Split[#]&]], {n, 0, 15}]
b[n_, lst_, v_] := b[n, lst, v] = If[n == 0, 1, If[n <= lst, 0, Sum[If[k == v, 0, b[n - k pz, pz, k]], {pz, lst + 1, n}, {k, Floor[n/pz]}]]]; a[n_] := b[n, 0, 0]; a /@ Range[0, 50] (* Giovanni Resta, May 18 2020 *)
CROSSREFS
Strictly increasing compositions are A000009. Partitions with strictly increasing run-lengths are A100471. Partitions with strictly decreasing run-lengths are A100881. Compositions with equal run-lengths are A329738. Compositions whose run-lengths are unimodal are A332726. Compositions with strictly increasing or decreasing run-lengths are A333191. Numbers with strictly increasing prime multiplicities are A334965. Cf. A072706, A098859, A100882, A100883, A304686, A329744, A329766, A332726, A332833, A332834, A332835, A333147, A333149, A333190.
Heinz numbers of integer partitions with strictly increasing first quotients.
+10
5
1, 2, 3, 4, 5, 6, 7, 9, 10, 11, 12, 13, 14, 15, 17, 19, 20, 21, 22, 23, 25, 26, 28, 29, 31, 33, 34, 35, 37, 38, 39, 41, 43, 44, 45, 46, 47, 49, 51, 52, 53, 55, 57, 58, 59, 61, 62, 63, 65, 66, 67, 68, 69, 71, 73, 74, 76, 77, 78, 79, 82, 83, 85, 86, 87, 89, 91
COMMENTS
The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). This gives a bijective correspondence between positive integers and integer partitions.
The first quotients of a sequence are defined as if the sequence were an increasing divisor chain, so for example the first quotients of (6,3,1) are (1/2,1/3).
EXAMPLE
The prime indices of 84 are {1,1,2,4}, with first quotients (1,2,2), so 84 is not in the sequence.
Most small numbers are in the sequence, but the sequence of non-terms together with their prime indices begins:
8: {1,1,1}
16: {1,1,1,1}
18: {1,2,2}
24: {1,1,1,2}
27: {2,2,2}
30: {1,2,3}
32: {1,1,1,1,1}
36: {1,1,2,2}
40: {1,1,1,3}
42: {1,2,4}
48: {1,1,1,1,2}
50: {1,3,3}
54: {1,2,2,2}
56: {1,1,1,4}
60: {1,1,2,3}
64: {1,1,1,1,1,1}
MATHEMATICA
primeptn[n_]:=If[n==1, {}, Reverse[Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]]];
Select[Range[100], Less@@Divide@@@Reverse/@Partition[primeptn[#], 2, 1]&]
CROSSREFS
For differences instead of quotients we have A325456 (count: A240027). For multiplicities (prime signature) instead of quotients we have A334965. The version counting strict divisor chains is A342086. The weakly increasing version is A342523. The strictly decreasing version is A342525. A167865 counts strict chains of divisors > 1 summing to n.
A318991/ A318992 rank reversed partitions with/without integer quotients.
A342098 counts (strict) partitions with all adjacent parts x > 2y.
Cf. A048767, A056239, A112798, A124010, A130091, A169594, A253249, A325351, A325352, A334997, A342530.
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