Search: a362619 -id:a362619
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A237824
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Number of partitions of n such that 2*(least part) >= greatest part.
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+10
29
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1, 2, 3, 4, 5, 7, 7, 10, 11, 13, 14, 19, 18, 23, 25, 29, 30, 38, 37, 46, 48, 54, 57, 70, 69, 80, 85, 97, 100, 118, 118, 137, 144, 159, 168, 193, 195, 220, 233, 259, 268, 303, 311, 348, 367, 399, 419, 469, 483, 532, 560, 610, 639, 704, 732, 801, 841, 908, 954
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OFFSET
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1,2
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COMMENTS
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By conjugation, also the number of integer partitions of n whose greatest part appears at a middle position, namely at k/2, (k+1)/2, or (k+2)/2 where k is the number of parts. These partitions have ranks A362622. - Gus Wiseman, May 14 2023
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LINKS
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FORMULA
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G.f.: Sum_{m>0} x^m/(1-x^m) + Sum_{i>0} Sum_{j=1..i} x^((2*i)+j) / Product_{k=0..j} (1 - x^(k+i)). - John Tyler Rascoe, Mar 07 2024
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EXAMPLE
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a(6) = 7 counts these partitions: 6, 42, 33, 222, 2211, 21111, 111111.
The a(1) = 1 through a(8) = 10 partitions such that 2*(least part) >= greatest part:
(1) (2) (3) (4) (5) (6) (7) (8)
(11) (21) (22) (32) (33) (43) (44)
(111) (211) (221) (42) (322) (53)
(1111) (2111) (222) (2221) (332)
(11111) (2211) (22111) (422)
(21111) (211111) (2222)
(111111) (1111111) (22211)
(221111)
(2111111)
(11111111)
The a(1) = 1 through a(8) = 10 partitions whose greatest part appears at a middle position:
(1) (2) (3) (4) (5) (6) (7) (8)
(11) (21) (22) (32) (33) (43) (44)
(111) (31) (41) (42) (52) (53)
(1111) (221) (51) (61) (62)
(11111) (222) (331) (71)
(2211) (2221) (332)
(111111) (1111111) (2222)
(3311)
(22211)
(11111111)
(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}] (* this sequence *)
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PROG
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(PARI)
N=60; x='x+O('x^N);
gf = sum(m=1, N, (x^m)/(1-x^m)) + sum(i=1, N, sum(j=1, i, x^((2*i)+j)/prod(k=0, j, 1 - x^(k+i))));
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CROSSREFS
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These partitions have ranks A362981.
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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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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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A362620
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Numbers whose greatest prime factor is not a mode, meaning it appears fewer times than some other.
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+10
7
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12, 20, 24, 28, 40, 44, 45, 48, 52, 56, 60, 63, 68, 72, 76, 80, 84, 88, 90, 92, 96, 99, 104, 112, 116, 117, 120, 124, 126, 132, 135, 136, 140, 144, 148, 152, 153, 156, 160, 164, 168, 171, 172, 175, 176, 180, 184, 188, 189, 192, 198, 200, 204, 207, 208, 212
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OFFSET
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1,1
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COMMENTS
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First differs from A112769 in lacking 300.
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LINKS
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EXAMPLE
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The prime factorization of 90 is 2*3*3*5, with modes {3} and maximum 5, so 90 is in the sequence.
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MAPLE
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filter:= proc(n) local F;
F:= sort(ifactors(n)[2], (a, b) -> a[1]<b[1]);
F[-1, 2] < max(F[.., 2])
end proc:
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MATHEMATICA
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prifacs[n_]:=If[n==1, {}, Flatten[ConstantArray@@@FactorInteger[n]]];
Select[Range[2, 100], FreeQ[Commonest[prifacs[#]], Max[prifacs[#]]]&]
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CROSSREFS
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Partitions of this type are counted by A240302.
A362605 ranks partitions with a more than one mode, counted by A362607.
A362606 ranks partitions with a more than one co-mode, counted by A362609.
A362611 counts modes in prime factorization, triangle version A362614.
A362613 counts co-modes in prime factorization, triangle version A362615.
A362621 ranks partitions with median equal to maximum, counted by A053263.
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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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A362980
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Numbers whose multiset of prime factors (with multiplicity) has different median from maximum.
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+10
4
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6, 10, 12, 14, 15, 20, 21, 22, 24, 26, 28, 30, 33, 34, 35, 36, 38, 39, 40, 42, 44, 45, 46, 48, 51, 52, 55, 56, 57, 58, 60, 62, 63, 65, 66, 68, 69, 70, 72, 74, 76, 77, 78, 80, 82, 84, 85, 86, 87, 88, 90, 91, 92, 93, 94, 95, 96, 99, 100, 102, 104, 105, 106, 110
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refs;
listen;
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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).
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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 not 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 in the sequence.
The terms together with their prime indices begin:
6: {1,2} 36: {1,1,2,2} 60: {1,1,2,3}
10: {1,3} 38: {1,8} 62: {1,11}
12: {1,1,2} 39: {2,6} 63: {2,2,4}
14: {1,4} 40: {1,1,1,3} 65: {3,6}
15: {2,3} 42: {1,2,4} 66: {1,2,5}
20: {1,1,3} 44: {1,1,5} 68: {1,1,7}
21: {2,4} 45: {2,2,3} 69: {2,9}
22: {1,5} 46: {1,9} 70: {1,3,4}
24: {1,1,1,2} 48: {1,1,1,1,2} 72: {1,1,1,2,2}
26: {1,6} 51: {2,7} 74: {1,12}
28: {1,1,4} 52: {1,1,6} 76: {1,1,8}
30: {1,2,3} 55: {3,5} 77: {4,5}
33: {2,5} 56: {1,1,1,4} 78: {1,2,6}
34: {1,7} 57: {2,8} 80: {1,1,1,1,3}
35: {3,4} 58: {1,10} 82: {1,13}
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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 A237821.
A362611 counts modes in prime factorization, triangle version A362614.
A362613 counts co-modes in prime factorization, triangle version A362615.
Cf. A000040, A002865, A171979, A237824, A237984, A327473, A327476, A359908, A362616, A362619, A362622.
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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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A362981
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Heinz numbers of integer partitions such that 2*(least part) >= greatest part.
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+10
3
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1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 12, 13, 15, 16, 17, 18, 19, 21, 23, 24, 25, 27, 29, 31, 32, 35, 36, 37, 41, 43, 45, 47, 48, 49, 53, 54, 55, 59, 61, 63, 64, 65, 67, 71, 72, 73, 75, 77, 79, 81, 83, 89, 91, 96, 97, 101, 103, 105, 107, 108, 109, 113, 119, 121, 125
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OFFSET
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1,2
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COMMENTS
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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.
By conjugation, also Heinz numbers of partitions whose greatest part appears at a middle position, namely k/2, (k+1)/2, or (k+2)/2, where k is the number of parts. These partitions have ranks A362622.
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LINKS
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EXAMPLE
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The terms together with their prime indices begin:
1: {} 16: {1,1,1,1} 36: {1,1,2,2}
2: {1} 17: {7} 37: {12}
3: {2} 18: {1,2,2} 41: {13}
4: {1,1} 19: {8} 43: {14}
5: {3} 21: {2,4} 45: {2,2,3}
6: {1,2} 23: {9} 47: {15}
7: {4} 24: {1,1,1,2} 48: {1,1,1,1,2}
8: {1,1,1} 25: {3,3} 49: {4,4}
9: {2,2} 27: {2,2,2} 53: {16}
11: {5} 29: {10} 54: {1,2,2,2}
12: {1,1,2} 31: {11} 55: {3,5}
13: {6} 32: {1,1,1,1,1} 59: {17}
15: {2,3} 35: {3,4} 61: {18}
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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[100], 2*Min@@prix[#]>=Max@@prix[#]&]
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CROSSREFS
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For prime factors instead of indices we have A081306.
Partitions of this type are counted by A237824.
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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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A362982
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Heinz numbers of partitions such that 2*(least part) < greatest part.
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+10
3
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10, 14, 20, 22, 26, 28, 30, 33, 34, 38, 39, 40, 42, 44, 46, 50, 51, 52, 56, 57, 58, 60, 62, 66, 68, 69, 70, 74, 76, 78, 80, 82, 84, 85, 86, 87, 88, 90, 92, 93, 94, 95, 98, 99, 100, 102, 104, 106, 110, 111, 112, 114, 115, 116, 117, 118, 120, 122, 123, 124, 126
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OFFSET
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1,1
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COMMENTS
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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.
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LINKS
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EXAMPLE
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The terms together with their prime indices begin:
10: {1,3} 44: {1,1,5} 70: {1,3,4}
14: {1,4} 46: {1,9} 74: {1,12}
20: {1,1,3} 50: {1,3,3} 76: {1,1,8}
22: {1,5} 51: {2,7} 78: {1,2,6}
26: {1,6} 52: {1,1,6} 80: {1,1,1,1,3}
28: {1,1,4} 56: {1,1,1,4} 82: {1,13}
30: {1,2,3} 57: {2,8} 84: {1,1,2,4}
33: {2,5} 58: {1,10} 85: {3,7}
34: {1,7} 60: {1,1,2,3} 86: {1,14}
38: {1,8} 62: {1,11} 87: {2,10}
39: {2,6} 66: {1,2,5} 88: {1,1,1,5}
40: {1,1,1,3} 68: {1,1,7} 90: {1,2,2,3}
42: {1,2,4} 69: {2,9} 92: {1,1,9}
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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[100], 2*Min@@prix[#]<Max@@prix[#]&]
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CROSSREFS
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For prime factors instead of indices we have A069900, complement A081306.
Partitions of this type are counted by A237820.
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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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