A381543
Numbers > 1 whose greatest prime index (A061395), number of distinct prime factors (A001221), and greatest prime multiplicity (A051903) are all equal.
Original entry on oeis.org
2, 12, 18, 36, 120, 270, 360, 540, 600, 750, 1080, 1350, 1500, 1680, 1800, 2250, 2700, 3000, 4500, 5040, 5400, 5670, 6750, 8400, 9000, 11340, 11760, 13500, 15120, 22680, 25200, 26250, 27000, 28350, 35280, 36960, 39690, 42000, 45360, 52500, 56700, 58800, 72030
Offset: 1
The terms together with their prime indices begin:
2: {1}
12: {1,1,2}
18: {1,2,2}
36: {1,1,2,2}
120: {1,1,1,2,3}
270: {1,2,2,2,3}
360: {1,1,1,2,2,3}
540: {1,1,2,2,2,3}
600: {1,1,1,2,3,3}
750: {1,2,3,3,3}
1080: {1,1,1,2,2,2,3}
1350: {1,2,2,2,3,3}
1500: {1,1,2,3,3,3}
1680: {1,1,1,1,2,3,4}
1800: {1,1,1,2,2,3,3}
Partitions of this type are counted by
A382302.
A122111 represents partition conjugation in terms of Heinz numbers.
-
Select[Range[2,1000],PrimePi[FactorInteger[#][[-1,1]]]==PrimeNu[#]==Max@@FactorInteger[#][[All,2]]&]
A101709
Number of partitions of n having nonnegative even rank (the rank of a partition is the largest part minus the number of parts).
Original entry on oeis.org
1, 0, 2, 1, 3, 2, 7, 5, 11, 10, 20, 20, 34, 35, 57, 62, 92, 104, 151, 171, 237, 274, 371, 433, 571, 670, 870, 1025, 1306, 1543, 1947, 2299, 2864, 3387, 4183, 4943, 6052, 7143, 8688, 10242, 12371, 14566, 17503, 20567, 24583, 28841, 34319, 40188, 47618, 55654, 65700, 76643, 90149, 104968
Offset: 1
a(5)=3 because the partitions of 5 with nonnegative even ranks are 5 (rank=4), 41 (rank=2) and 311 (rank=0).
- George E. Andrews, The Theory of Partitions, Addison-Wesley, Reading, Mass., 1976.
A340611
Number of integer partitions of n of length 2^k where k is the greatest part.
Original entry on oeis.org
1, 0, 1, 0, 0, 1, 1, 1, 1, 0, 1, 1, 2, 2, 3, 3, 4, 4, 4, 4, 4, 4, 5, 5, 6, 7, 8, 10, 12, 14, 16, 19, 21, 24, 27, 29, 32, 34, 36, 38, 41, 42, 45, 47, 50, 52, 56, 58, 63, 66, 71, 75, 83, 88, 98, 106, 118, 128, 143, 155, 173, 188, 208, 226, 250, 270, 297, 321, 350
Offset: 0
The partitions for n = 12, 14, 16, 22, 24:
32211111 32222111 32222221 33333322 33333333
33111111 33221111 33222211 33333331 4222221111111111
33311111 33322111 4222111111111111 4322211111111111
33331111 4321111111111111 4332111111111111
4411111111111111 4422111111111111
4431111111111111
The conjugate partitions:
(8,2,2) (8,3,3) (8,4,4) (8,7,7) (8,8,8)
(8,3,1) (8,4,2) (8,5,3) (8,8,6) (16,3,3,2)
(8,5,1) (8,6,2) (16,2,2,2) (16,4,2,2)
(8,7,1) (16,3,2,1) (16,4,3,1)
(16,4,1,1) (16,5,2,1)
(16,6,1,1)
Note: A-numbers of Heinz-number sequences are in parentheses below.
A072233 counts partitions by sum and length.
A168659 = partitions whose greatest part divides their length (
A340609).
A168659 = partitions whose length divides their greatest part (
A340610).
A326843 = partitions of n whose length and maximum both divide n (
A326837).
A340597 lists numbers with an alt-balanced factorization.
A340653 counts balanced factorizations.
A340689 have a factorization of length 2^max.
A340690 have a factorization of maximum 2^length.
-
Table[Length[Select[IntegerPartitions[n],Length[#]==2^Max@@#&]],{n,0,30}]
A340689
Numbers with a factorization of length 2^k into factors > 1, where k is the greatest factor.
Original entry on oeis.org
1, 16, 384, 576, 864, 1296, 1944, 2916, 4374, 6561, 131072, 196608, 262144, 294912, 393216, 442368, 524288, 589824, 663552, 786432, 884736, 995328, 1048576, 1179648, 1327104, 1492992, 1572864, 1769472, 1990656, 2097152, 2239488, 2359296, 2654208, 2985984, 3145728
Offset: 1
The initial terms and a valid factorization of each are:
1 =
16 = 2*2*2*2
384 = 2*2*2*2*2*2*2*3
576 = 2*2*2*2*2*2*3*3
864 = 2*2*2*2*2*3*3*3
1296 = 2*2*2*2*3*3*3*3
1944 = 2*2*2*3*3*3*3*3
2916 = 2*2*3*3*3*3*3*3
4374 = 2*3*3*3*3*3*3*3
6561 = 3*3*3*3*3*3*3*3
131072 = 2*2*2*2*2*2*2*2*2*2*2*2*2*2*2*4
196608 = 2*2*2*2*2*2*2*2*2*2*2*2*2*2*3*4
262144 = 2*2*2*2*2*2*2*2*2*2*2*2*2*2*4*4
294912 = 2*2*2*2*2*2*2*2*2*2*2*2*2*3*3*4
Partitions of the prescribed type are counted by
A340611.
A047993 counts balanced partitions.
A316439 counts factorizations by product and length.
A340596 counts co-balanced factorizations.
A340597 lists numbers with an alt-balanced factorization.
A340653 counts balanced factorizations.
Cf.
A106529,
A117409,
A200750,
A325134,
A340386,
A340387,
A340599,
A340607,
A340654,
A340655,
A340656,
A340657.
-
facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
Select[Range[1000],Select[facs[#],Length[#]==2^Max@@#&]!={}&]
A340856
Squarefree numbers whose greatest prime index (A061395) is divisible by their number of prime factors (A001222).
Original entry on oeis.org
2, 3, 5, 6, 7, 11, 13, 14, 17, 19, 21, 23, 26, 29, 30, 31, 35, 37, 38, 39, 41, 43, 47, 53, 57, 58, 59, 61, 65, 67, 71, 73, 74, 78, 79, 83, 86, 87, 89, 91, 95, 97, 101, 103, 106, 107, 109, 111, 113, 122, 127, 129, 130, 131, 133, 137, 138, 139, 142, 143, 145
Offset: 1
The sequence of terms together with their prime indices begins:
2: {1} 31: {11} 71: {20}
3: {2} 35: {3,4} 73: {21}
5: {3} 37: {12} 74: {1,12}
6: {1,2} 38: {1,8} 78: {1,2,6}
7: {4} 39: {2,6} 79: {22}
11: {5} 41: {13} 83: {23}
13: {6} 43: {14} 86: {1,14}
14: {1,4} 47: {15} 87: {2,10}
17: {7} 53: {16} 89: {24}
19: {8} 57: {2,8} 91: {4,6}
21: {2,4} 58: {1,10} 95: {3,8}
23: {9} 59: {17} 97: {25}
26: {1,6} 61: {18} 101: {26}
29: {10} 65: {3,6} 103: {27}
30: {1,2,3} 67: {19} 106: {1,16}
Note: Heinz number sequences are given in parentheses below.
The case of equality, and the reciprocal version, are both
A002110.
These are the Heinz numbers of partitions counted by
A340828.
A006141 counts partitions whose length equals their minimum (
A324522).
A061395 selects the maximum prime index.
A112798 lists the prime indices of each positive integer.
A257541 gives the rank of the partition with Heinz number n.
A340830 counts strict partitions whose parts are multiples of the length.
-
Select[Range[2,100],SquareFreeQ[#]&&Divisible[PrimePi[FactorInteger[#][[-1,1]]],PrimeOmega[#]]&]
A349150
Heinz numbers of integer partitions with at most one odd part.
Original entry on oeis.org
1, 2, 3, 5, 6, 7, 9, 11, 13, 14, 15, 17, 18, 19, 21, 23, 26, 27, 29, 31, 33, 35, 37, 38, 39, 41, 42, 43, 45, 47, 49, 51, 53, 54, 57, 58, 59, 61, 63, 65, 67, 69, 71, 73, 74, 77, 78, 79, 81, 83, 86, 87, 89, 91, 93, 95, 97, 98, 99, 101, 103, 105, 106, 107, 109
Offset: 1
The terms and their prime indices begin:
1: {} 23: {9} 49: {4,4}
2: {1} 26: {1,6} 51: {2,7}
3: {2} 27: {2,2,2} 53: {16}
5: {3} 29: {10} 54: {1,2,2,2}
6: {1,2} 31: {11} 57: {2,8}
7: {4} 33: {2,5} 58: {1,10}
9: {2,2} 35: {3,4} 59: {17}
11: {5} 37: {12} 61: {18}
13: {6} 38: {1,8} 63: {2,2,4}
14: {1,4} 39: {2,6} 65: {3,6}
15: {2,3} 41: {13} 67: {19}
17: {7} 42: {1,2,4} 69: {2,9}
18: {1,2,2} 43: {14} 71: {20}
19: {8} 45: {2,2,3} 73: {21}
21: {2,4} 47: {15} 74: {1,12}
These are the positions of 0's and 1's in
A257991.
The conjugate partitions are ranked by
A349151.
A122111 is a representation of partition conjugation.
A300063 ranks partitions of odd numbers, counted by
A058695 up to 0's.
A316524 gives the alternating sum of prime indices (reverse:
A344616).
A325698 ranks partitions with as many even as odd parts, counted by
A045931.
A345958 ranks partitions with alternating sum 1.
A349157 ranks partitions with as many even parts as odd conjugate parts.
Cf.
A000290,
A000700,
A001222,
A027187,
A027193,
A028260,
A035363,
A047993,
A215366,
A257992,
A277579,
A326841.
-
primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],Count[Reverse[primeMS[#]],_?OddQ]<=1&]
A381079
Number of integer partitions of n whose greatest multiplicity is equal to their sum of distinct parts.
Original entry on oeis.org
0, 1, 0, 0, 1, 1, 0, 3, 1, 3, 1, 2, 0, 7, 2, 6, 7, 11, 3, 19, 8, 22, 16, 32, 17, 48, 21, 50, 39, 71, 35, 101, 58, 120, 89, 156, 97, 228, 133, 267, 203, 352, 228, 483, 322, 571, 444, 734, 524, 989, 683, 1160, 942, 1490, 1103, 1919, 1438, 2302, 1890, 2881, 2243, 3683, 2842, 4384, 3703, 5461
Offset: 0
The partition (3,2,2,1,1,1,1,1,1) has greatest multiplicity 6 and distinct parts (3,2,1) with sum 6, so is counted under a(13).
The a(1) = 1 through a(13) = 7 partitions:
1 . . 22 2111 . 2221 22211 333 331111 5111111 . 33331
22111 222111 32111111 322222
31111 411111 3331111
4411111
61111111
322111111
421111111
For greatest part instead of multiplicity we have
A000005.
These partitions are ranked by
A381632, for part instead of multiplicity
A246655.
A091605 counts partitions with greatest multiplicity 2.
A240312 counts partitions with max part = max multiplicity, ranks
A381542.
-
Table[Length[Select[IntegerPartitions[n],Max@@Length/@Split[#]==Total[Union[#]]&]],{n,0,30}]
A381544
Number of integer partitions of n not containing more ones than any other part.
Original entry on oeis.org
0, 0, 1, 2, 3, 4, 7, 8, 13, 17, 24, 30, 45, 54, 75, 97, 127, 160, 212, 263, 342, 427, 541, 672, 851, 1046, 1307, 1607, 1989, 2428, 2993, 3631, 4443, 5378, 6533, 7873, 9527, 11424, 13752, 16447, 19701, 23470, 28016, 33253, 39537, 46801, 55428, 65408, 77238
Offset: 0
The a(2) = 1 through a(9) = 17 partitions:
(2) (3) (4) (5) (6) (7) (8) (9)
(21) (22) (32) (33) (43) (44) (54)
(31) (41) (42) (52) (53) (63)
(221) (51) (61) (62) (72)
(222) (322) (71) (81)
(321) (331) (332) (333)
(2211) (421) (422) (432)
(2221) (431) (441)
(521) (522)
(2222) (531)
(3221) (621)
(3311) (3222)
(22211) (3321)
(4221)
(22221)
(32211)
(222111)
The Heinz numbers of these partitions are
A381439.
A091602 counts partitions by the greatest multiplicity, rank statistic
A051903.
A239964 counts partitions with max multiplicity = length, ranks
A212166.
A240312 counts partitions with max part = max multiplicity, ranks
A381542.
A382302 counts partitions with max = max multiplicity = distinct length, ranks
A381543.
-
Table[Length[Select[IntegerPartitions[n],Count[#,1]<=Max@@Length/@Split[DeleteCases[#,1]]&]],{n,0,30}]
A384888
Number of integer partitions of n with all equal lengths of maximal anti-runs (decreasing by more than 1).
Original entry on oeis.org
1, 1, 2, 3, 5, 6, 9, 10, 13, 17, 20, 24, 32, 36, 44, 55, 64, 75, 92, 105, 125, 147, 169, 195, 231, 263, 303, 351, 401, 458, 532, 600, 686, 784, 889, 1010, 1152, 1296, 1468, 1662, 1875, 2108, 2384, 2669, 3001, 3373, 3775, 4222, 4734, 5278, 5896, 6576, 7322
Offset: 0
The partition y = (10,6,6,4,3,1) has maximal anti-runs ((10,6),(6,4),(3,1)), with lengths (2,2,2), so y is counted under a(30).
The a(1) = 1 through a(8) = 13 partitions:
(1) (2) (3) (4) (5) (6) (7) (8)
(11) (21) (22) (32) (33) (43) (44)
(111) (31) (41) (42) (52) (53)
(211) (221) (51) (61) (62)
(1111) (2111) (222) (322) (71)
(11111) (321) (2221) (332)
(2211) (3211) (2222)
(21111) (22111) (3221)
(111111) (211111) (22211)
(1111111) (32111)
(221111)
(2111111)
(11111111)
The strict case is new, distinct
A384880.
For distinct instead of equal lengths we have
A384885.
A098859 counts Wilf partitions (distinct multiplicities), complement
A336866.
A355394 counts partitions without a neighborless part, singleton case
A355393.
A356236 counts partitions with a neighborless part, singleton case
A356235.
A356606 counts strict partitions without a neighborless part, complement
A356607.
Cf.
A008284,
A044813,
A047993,
A242882,
A287170,
A325325,
A356226,
A384175,
A384176,
A384178,
A384886.
-
Table[Length[Select[IntegerPartitions[n],SameQ@@Length/@Split[#,#2<#1-1&]&]],{n,0,15}]
A096419
Number of cyclically symmetric plane partitions (Macdonald's plane partition conjecture).
Original entry on oeis.org
1, 0, 0, 1, 0, 0, 2, 1, 0, 2, 1, 0, 4, 3, 0, 5, 4, 0, 8, 8, 0, 10, 11, 0, 15, 19, 1, 20, 27, 1, 28, 43, 3, 36, 61, 6, 50, 92, 11, 64, 129, 18, 86, 189, 33, 110, 262, 51, 145, 374, 84, 184, 514, 129, 238, 718, 201, 300, 977, 300, 384, 1344, 454, 482, 1812, 661, 609, 2459, 972
Offset: 1
- Andrews, G. E. "Plane Partitions (III): The Weak Macdonald Conjecture." Invent. Math. 53, 193-225, 1979.
- Mills, W. H.; Robbins, D. P.; and Rumsey, H. Jr., Proof of the Macdonald Conjecture. Invent. Math. 66, 73-87, 1982.
-
len=151;m=Ceiling[len/3];mcdon=Rest@CoefficientList[Series[Product[(1-q^(3i-1))/(1-q^(3i-2)) Product[(1-q^(3(m+i+j-1)))/(1-q^(3(2i+j-1))), {j, i, m}], {i, 1, m}], {q, 0, len}], q] (* updated by Wouter Meeussen, Apr 15 2025 *)
Comments