Search: a362980 -id:a362980
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A237821
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Number of partitions of n such that 2*(least part) <= greatest part.
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+10
17
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0, 0, 1, 2, 4, 7, 11, 16, 25, 35, 48, 68, 92, 123, 164, 216, 282, 367, 471, 604, 769, 975, 1225, 1542, 1924, 2395, 2968, 3669, 4514, 5547, 6781, 8280, 10071, 12229, 14796, 17881, 21537, 25902, 31066, 37206, 44443, 53021, 63098, 74995, 88946, 105350, 124533
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OFFSET
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1,4
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COMMENTS
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By conjugation, also the number of integer partitions of n with different median from maximum, ranks A362980. - Gus Wiseman, May 15 2023
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LINKS
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FORMULA
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G.f.: Sum_{i>=1} Sum_{j>=0} x^(3*i+j) /Product_{k=i..2*i+j} (1-x^k). - Seiichi Manyama, May 27 2023
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EXAMPLE
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a(6) = 7 counts these partitions: 51, 42, 411, 321, 3111, 2211, 21111.
The a(3) = 1 through a(8) = 16 partitions wirth 2*(least part) <= greatest part:
(21) (31) (41) (42) (52)
(211) (221) (51) (61)
(311) (321) (331)
(2111) (411) (421)
(2211) (511)
(3111) (2221)
(21111) (3211)
(4111)
(22111)
(31111)
(211111)
The a(3) = 1 through a(8) = 16 partitions with different median from maximum:
(21) (31) (32) (42) (43)
(211) (41) (51) (52)
(311) (321) (61)
(2111) (411) (322)
(2211) (421)
(3111) (511)
(21111) (3211)
(4111)
(22111)
(31111)
(211111)
(End)
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MATHEMATICA
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z = 60; q[n_] := q[n] = IntegerPartitions[n];
Table[Count[q[n], p_ /; 2 Min[p] < Max[p]], {n, z}] (* A237820 *)
Table[Count[q[n], p_ /; 2 Min[p] <= Max[p]], {n, z}] (* A237821 *)
Table[Count[q[n], p_ /; 2 Min[p] = = Max[p]], {n, z}](* A118096 *)
Table[Count[q[n], p_ /; 2 Min[p] > Max[p]], {n, z}] (* A053263 *)
Table[Count[q[n], p_ /; 2 Min[p] >= Max[p]], {n, z}] (* A237824 *)
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CROSSREFS
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These partitions have ranks A069900.
The conjugate partitions have ranks A362980.
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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A362621
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One and numbers whose multiset of prime factors (with multiplicity) has the same median as maximum.
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+10
10
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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
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OFFSET
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1,2
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COMMENTS
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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).
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LINKS
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EXAMPLE
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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}
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MATHEMATICA
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Select[Range[100], (y=Flatten[Apply[ConstantArray, FactorInteger[#], {1}]]; Max@@y==Median[y])&]
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CROSSREFS
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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.
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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A362622
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One and numbers whose prime factorization has its greatest part at a middle position.
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+10
9
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1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 14, 15, 16, 17, 18, 19, 21, 22, 23, 25, 26, 27, 29, 31, 32, 33, 34, 35, 36, 37, 38, 39, 41, 43, 46, 47, 49, 50, 51, 53, 54, 55, 57, 58, 59, 61, 62, 64, 65, 67, 69, 71, 73, 74, 75, 77, 79, 81, 82, 83, 85, 86, 87, 89, 91
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OFFSET
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1,2
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LINKS
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EXAMPLE
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The prime factorization of 150 is 5*5*3*2, with middle parts {3,5}, so 150 is in the sequence.
The prime factorization of 90 is 5*3*3*2, with middle parts {3,3}, so 90 is not in the sequence.
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MATHEMATICA
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mpm[q_]:=MemberQ[If[OddQ[Length[q]], {Median[q]}, {q[[Length[q]/2]], q[[Length[q]/2+1]]}], Max@@q];
Select[Range[100], #==1||mpm[Flatten[Apply[ConstantArray, FactorInteger[#], {1}]]]&]
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CROSSREFS
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Partitions of this type are counted by A237824.
The version for median instead of middles is A362621, counted by A053263.
A362611 counts modes in prime factorization.
A362613 counts co-modes in prime factorization.
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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A363223
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Numbers with bigomega equal to median prime index.
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+10
2
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2, 9, 10, 50, 70, 75, 105, 110, 125, 130, 165, 170, 175, 190, 195, 230, 255, 275, 285, 290, 310, 325, 345, 370, 410, 425, 430, 435, 465, 470, 475, 530, 555, 575, 590, 610, 615, 645, 670, 686, 705, 710, 725, 730, 775, 790, 795, 830, 885, 890, 915, 925, 970
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OFFSET
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1,1
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COMMENTS
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The median of a multiset is either the middle part (for odd length), or the average of the two middle parts (for even length).
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.
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LINKS
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FORMULA
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EXAMPLE
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The terms together with their prime indices begin:
2: {1}
9: {2,2}
10: {1,3}
50: {1,3,3}
70: {1,3,4}
75: {2,3,3}
105: {2,3,4}
110: {1,3,5}
125: {3,3,3}
130: {1,3,6}
165: {2,3,5}
170: {1,3,7}
175: {3,3,4}
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MATHEMATICA
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prix[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];
Select[Range[1000], PrimeOmega[#]==Median[prix[#]]&]
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CROSSREFS
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Partitions of this type are counted by A361800.
A000975 counts subsets with integer median.
A359908 lists numbers whose prime indices have integer median.
A360005 gives twice median of prime indices.
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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