A064173 -id:A064173 - cp's OEIS frontend

A340605 Heinz numbers of integer partitions of even positive rank.

5, 11, 14, 17, 21, 23, 26, 31, 35, 38, 39, 41, 44, 47, 49, 57, 58, 59, 65, 66, 67, 68, 73, 74, 83, 86, 87, 91, 92, 95, 97, 99, 102, 103, 104, 106, 109, 110, 111, 122, 124, 127, 129, 133, 137, 138, 142, 143, 145, 149, 152, 153, 154, 156, 157, 158, 159, 164, 165

Offset: 1

Gus Wiseman, Jan 21 2021

nonn

The Dyson rank of a nonempty partition is its maximum part minus its number of parts. The rank of an empty partition is 0.

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 sequence of partitions with their Heinz numbers begins:
      5: (3)         57: (8,2)       97: (25)
     11: (5)         58: (10,1)      99: (5,2,2)
     14: (4,1)       59: (17)       102: (7,2,1)
     17: (7)         65: (6,3)      103: (27)
     21: (4,2)       66: (5,2,1)    104: (6,1,1,1)
     23: (9)         67: (19)       106: (16,1)
     26: (6,1)       68: (7,1,1)    109: (29)
     31: (11)        73: (21)       110: (5,3,1)
     35: (4,3)       74: (12,1)     111: (12,2)
     38: (8,1)       83: (23)       122: (18,1)
     39: (6,2)       86: (14,1)     124: (11,1,1)
     41: (13)        87: (10,2)     127: (31)
     44: (5,1,1)     91: (6,4)      129: (14,2)
     47: (15)        92: (9,1,1)    133: (8,4)
     49: (4,4)       95: (8,3)      137: (33)

FindStat, St000145: The Dyson rank of a partition

Note: Heinz numbers are given in parentheses below.
Allowing any positive rank gives A064173 (A340787).
The odd version is counted by A101707 (A340604).
These partitions are counted by A101708.
The not necessarily positive case is counted by A340601 (A340602).
A001222 counts prime indices.
A061395 gives maximum prime index.
A072233 counts partitions by sum and length.
- Rank -
A047993 counts partitions of rank 0 (A106529).
A064173 counts partitions of negative rank (A340788).
A064174 counts partitions of nonnegative rank (A324562).
A064174 (also) counts partitions of nonpositive rank (A324521).
A101198 counts partitions of rank 1 (A325233).
A257541 gives the rank of the partition with Heinz number n.
A340692 counts partitions of odd rank (A340603).
- Even -
A027187 counts partitions of even length (A028260).
A027187 (also) counts partitions of even maximum (A244990).
A035363 counts partitions into even parts (A066207).
A058696 counts partitions of even numbers (A300061).
A067661 counts strict partitions of even length (A030229).
A339846 counts factorizations of even length.
Cf. A006141, A024430, A056239, A112798, A340387, A340598, A340600, A340608, A340609, A340610, A340653.

rk[n_]:=PrimePi[FactorInteger[n][[-1,1]]]-PrimeOmega[n];
Select[Range[100],EvenQ[rk[#]]&&rk[#]>0&]

A061395(a(n)) - A001222(a(n)) is even and positive.

A340604 \/ A340605 = A340787.

A208478 Triangle read by rows: T(n,k) = number of partitions of n with positive k-th rank.

Original entry on oeis.org

0, 1, 1, 1, 1, 1, 2, 1, 2, 1, 3, 1, 3, 2, 1, 5, 2, 4, 4, 2, 1, 6, 3, 5, 6, 4, 2, 1, 10, 5, 7, 9, 7, 4, 2, 1, 13, 7, 9, 11, 11, 7, 4, 2, 1, 19, 11, 12, 15, 16, 12, 7, 4, 2, 1, 25, 16, 15, 19, 22, 18, 12, 7, 4, 2, 1, 35, 24, 20, 26, 29, 27, 19, 12, 7, 4, 2, 1

Offset: 1

Omar E. Pol, Mar 07 2012

We define the k-th rank of a partition as the k-th part minus the number of parts >= k. Every partition of n has n ranks. This is a generalization of the Dyson's rank of a partition which is the largest part minus the number of parts. Since the first part of a partition is also the largest part of the same partition so the Dyson's rank of a partition is the case for k = 1.
The sum of the k-th ranks of all partitions of n is equal to zero.
Also T(n,k) = number of partitions of n with negative k-th rank.
It appears that reversed rows converge to A000070, the same as A208482. - Omar E. Pol, Mar 11 2012
From Omar E. Pol, Dec 12 2019: (Start)
1) The k-th part of a partition of n is also the number of parts >= k of its conjugate partition.
2) The k-th rank of a partitions is also the number of parts >= k of its conjugate partition minus the number of parts >= k.
For example: for n = 9 consider the partition [5, 3, 1]. The first part is 5, so the conjugate partition has five parts >= 1. The second part is 3, so the conjugate partition has three parts >= 2. The third part is 1, so the conjugate partition has only one part >= 3. The mentioned conjugate partition is [3, 2, 2, 1, 1]. And conversely, consider the partition [3, 2, 2, 1, 1]. The first part is 3, so the conjugate partition has three parts >= 1. The second part is 2, so the conjugate partition has two parts >= 2. the Third part is 2, so the conjugate partition has two parts >= 3, and so on. In this case the conjugate partition is [5, 3, 1].
3) The difference between the k-th part and the (k+1)-st part of the partition of n is also the number of k's in its conjugate partition. For example: consider the partition [5, 3, 1]. The difference between the first and the second part is 5 - 3 = 2, equals the number of 1's in its conjugate partition. The difference between the second and the third part is 3 - 1 = 2, equals the number of 2's in its conjugate partition. The difference between the third and the fourth (virtual) part is 1 - 0 = 1, equals the number of 3's in its conjugate partition [3, 2, 2, 1, 1]. And conversely, consider the partition [3, 2, 2, 1, 1]. The difference between the first and the second part is 3 - 2 = 1, equals the number of 1's in its conjugate partition. The difference between the second and the third part is 2 - 2 = 0, equals the number of 2's in its conjugate partition. The difference between the third and the fourth part is 2 - 1 = 1, equals the number of 3's in its conjugate partition, and so on.
4) The list of n ranks of a partition of n equals the list of n ranks multiplied by -1 of its conjugate partition. For example the nine ranks of the partition [5, 3, 1] of 9 are [2, 1, -1, -1, -1, -1, 0, 0, 0], and the nine ranks of its conjugate partition [3, 2, 2, 1, 1] are [-2, -1, 1, 1, 1, 1, 0, 0, 0].
For a list of partitions of the positive integers ordered by its k-th ranks see A330370. (End)

			For n = 4 the partitions of 4 and the four types of ranks of the partitions of 4 are
----------------------------------------------------------
Partitions    First      Second       Third      Fourth
of 4          rank        rank        rank        rank
----------------------------------------------------------
4           4-1 =  3    0-1 = -1    0-1 = -1    0-1 = -1
3+1         3-2 =  1    1-1 =  0    0-1 = -1    0-0 =  0
2+2         2-2 =  0    2-2 =  0    0-0 =  0    0-0 =  0
2+1+1       2-3 = -1    1-1 =  0    1-0 =  1    0-0 =  0
1+1+1+1     1-4 = -3    1-0 =  1    1-0 =  1    1-0 =  1
----------------------------------------------------------
The number of partitions of 4 with positive k-th ranks are 2, 1, 2, 1 so row 4 lists 2, 1, 2, 1.
Triangle begins:
   0;
   1,  1;
   1,  1,  1;
   2,  1,  2,  1;
   3,  1,  3,  2,  1;
   5,  2,  4,  4,  2,  1;
   6,  3,  5,  6,  4,  2,  1;
  10,  5,  7,  9,  7,  4,  2,  1;
  13,  7,  9, 11, 11,  7,  4,  2,  1;
  19, 11, 12, 15, 16, 12,  7,  4,  2,  1;
  25, 16, 15, 19, 22, 18, 12,  7,  4,  2,  1;
  35, 24, 20, 26, 29, 27, 19, 12,  7,  4,  2,  1;
  ...

Alois P. Heinz, Rows n = 1..44, flattened

Column 1 is A064173.
Row sums give A208479.
Cf. A063995, A105805, A181187, A194547, A194549, A195822, A208482, A208483, A209616, A330368, A330369, A330370.

More terms from Alois P. Heinz, Mar 11 2012

A340787 Heinz numbers of integer partitions of positive rank.

Original entry on oeis.org

3, 5, 7, 10, 11, 13, 14, 15, 17, 19, 21, 22, 23, 25, 26, 28, 29, 31, 33, 34, 35, 37, 38, 39, 41, 42, 43, 44, 46, 47, 49, 51, 52, 53, 55, 57, 58, 59, 61, 62, 63, 65, 66, 67, 68, 69, 70, 71, 73, 74, 76, 77, 78, 79, 82, 83, 85, 86, 87, 88, 89, 91, 92, 93, 94, 95

Offset: 1

Gus Wiseman, Jan 29 2021

nonn

The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k), giving a bijective correspondence between positive integers and integer partitions.

The Dyson rank of a nonempty partition is its maximum part minus its length. The rank of an empty partition is undefined.

			The sequence of partitions together with their Heinz numbers begins:
     3: (2)      28: (4,1,1)    49: (4,4)      69: (9,2)
     5: (3)      29: (10)       51: (7,2)      70: (4,3,1)
     7: (4)      31: (11)       52: (6,1,1)    71: (20)
    10: (3,1)    33: (5,2)      53: (16)       73: (21)
    11: (5)      34: (7,1)      55: (5,3)      74: (12,1)
    13: (6)      35: (4,3)      57: (8,2)      76: (8,1,1)
    14: (4,1)    37: (12)       58: (10,1)     77: (5,4)
    15: (3,2)    38: (8,1)      59: (17)       78: (6,2,1)
    17: (7)      39: (6,2)      61: (18)       79: (22)
    19: (8)      41: (13)       62: (11,1)     82: (13,1)
    21: (4,2)    42: (4,2,1)    63: (4,2,2)    83: (23)
    22: (5,1)    43: (14)       65: (6,3)      85: (7,3)
    23: (9)      44: (5,1,1)    66: (5,2,1)    86: (14,1)
    25: (3,3)    46: (9,1)      67: (19)       87: (10,2)
    26: (6,1)    47: (15)       68: (7,1,1)    88: (5,1,1,1)

Freeman J. Dyson, A new symmetry of partitions, Journal of Combinatorial Theory 7.1 (1969): 56-61.
FindStat, St000145: The Dyson rank of a partition

Note: A-numbers of Heinz-number sequences are in parentheses below.
These partitions are counted by A064173.
The odd case is A101707 (A340604).
The even case is A101708 (A340605).
The negative version is (A340788).
A001222 counts prime factors.
A061395 selects the maximum prime index.
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).
A200750 = partitions whose length and maximum are relatively prime.
- Rank -
A047993 counts partitions of rank 0 (A106529).
A063995/A105806 count partitions by Dyson rank.
A064174 counts partitions of nonnegative/nonpositive rank (A324562/A324521).
A101198 counts partitions of rank 1 (A325233).
A257541 gives the rank of the partition with Heinz number n.
A324520 counts partitions with rank equal to least part (A324519).
A340601 counts partitions of even rank (A340602), with strict case A117192.
A340692 counts partitions of odd rank (A340603), with strict case A117193.
Cf. A003114, A006141, A039900, A056239, A096401, A112798, A117409, A316413, A324517, A325134, A326845, A340828.

Select[Range[2,100],PrimePi[FactorInteger[#][[-1,1]]]>PrimeOmega[#]&]

For all terms A061395(a(n)) > A001222(a(n)).

A340828 Number of strict integer partitions of n whose maximum part is a multiple of their length.

Original entry on oeis.org

1, 1, 2, 1, 2, 3, 3, 2, 4, 5, 6, 6, 7, 8, 11, 10, 13, 17, 18, 21, 24, 27, 30, 35, 39, 46, 53, 61, 68, 79, 87, 97, 110, 123, 139, 157, 175, 196, 222, 247, 278, 312, 347, 385, 433, 476, 531, 586, 651, 720, 800, 883, 979, 1085, 1200, 1325, 1464, 1614, 1777

Offset: 1

Gus Wiseman, Feb 01 2021

nonn

			The a(1) = 1 through a(16) = 10 partitions (A..G = 10..16):
  1  2  3   4  5   6    7   8   9    A     B    C    D    E     F      G
        21     41  42   43  62  63   64    65   84   85   86    87     A6
                   321  61      81   82    83   A2   A3   A4    A5     C4
                                621  631   A1   642  C1   C2    C3     E2
                                     4321  632  651  643  653   E1     943
                                           641  921  652  932   654    952
                                                     931  941   942    961
                                                          8321  951    C31
                                                                C21    8431
                                                                8421   8521
                                                                54321

FindStat, St000010: The length of the partition.
FindStat, St000147: The largest part of an integer partition.
FindStat, St000784: The maximum of the length and the largest part of the integer partition.

Note: A-numbers of Heinz-number sequences are in parentheses below.
The non-strict version is A168659 (A340609/A340610).
A018818 counts partitions into divisors (A326841).
A047993 counts balanced partitions (A106529).
A064173 counts partitions of positive/negative rank (A340787/A340788).
A067538 counts partitions whose length/max divides sum (A316413/A326836).
A072233 counts partitions by sum and length, with strict case A008289.
A096401 counts strict partition with length equal to minimum.
A102627 counts strict partitions with length dividing sum.
A326842 counts partitions whose length and parts all divide sum (A326847).
A326850 counts strict partitions whose maximum part divides sum.
A326851 counts strict partitions with length and maximum dividing sum.
A340829 counts strict partitions with Heinz number divisible by sum.
A340830 counts strict partitions with all parts divisible by length.
Cf. A003114, A006141, A064174, A117409, A143773 (A316428), A200750, A326843 (A326837), A330950 (A324851).

Table[Length[Select[IntegerPartitions[n],UnsameQ@@#&&Divisible[Max@@#,Length[#]]&]],{n,30}]

A208482 Triangle read by rows: T(n,k) = sum of positive k-th ranks of all partitions of n.

Original entry on oeis.org

0, 1, 1, 2, 1, 1, 4, 1, 2, 1, 7, 1, 3, 2, 1, 12, 2, 5, 4, 2, 1, 18, 3, 6, 6, 4, 2, 1, 29, 6, 9, 10, 7, 4, 2, 1, 42, 9, 11, 13, 11, 7, 4, 2, 1, 63, 16, 15, 19, 17, 12, 7, 4, 2, 1, 89, 24, 18, 25, 24, 18, 12, 7, 4, 2, 1, 128, 39, 24, 36, 34, 28, 19, 12, 7, 4, 2, 1

Offset: 1

Omar E. Pol, Mar 07 2012

For the definition of the k-th rank see A208478.
It appears that the sum of the k-th ranks of all partitions of n is equal to zero.
It appears that reversed rows converge to A000070, the same as A208478. - Omar E. Pol, Mar 10 2012

			For n = 4 the partitions of 4 and the four types of ranks of the partitions of 4 are
----------------------------------------------------------
Partitions    First      Second       Third      Fourth
of 4          rank        rank        rank        rank
----------------------------------------------------------
4           4-1 =  3    0-1 = -1    0-1 = -1    0-1 = -1
3+1         3-2 =  1    1-1 =  0    0-1 = -1    0-0 =  0
2+2         2-2 =  0    2-2 =  0    0-0 =  0    0-0 =  0
2+1+1       2-3 = -1    1-1 =  0    1-0 =  1    0-0 =  0
1+1+1+1     1-4 = -3    1-0 =  1    1-0 =  1    1-0 =  1
----------------------------------------------------------
The sums of positive k-th ranks of the partitions of 4 are 4, 1, 2, 1 so row 4 lists 4, 1, 2, 1.
Triangle begins:
0;
1,    1;
2,    1,  1;
4,    1,  2,  1;
7,    1,  3,  2,  1;
12,   2,  5,  4,  2,  1;
18,   3,  6,  6,  4,  2,  1;
29,   6,  9, 10,  7,  4,  2,  1;
42,   9, 11, 13, 11,  7,  4,  2,  1;
63,  16, 15, 19, 17, 12,  7,  4,  2,  1;
89,  24, 18, 25, 24, 18, 12,  7,  4,  2,  1;
128, 39, 24, 36, 34, 28, 19, 12,  7,  4,  2,  1;

Alois P. Heinz, Rows n = 1..44, flattened

Column 1 is A209616. Row sums give A208483.

Cf. A006128, A063995, A064173, A105805, A181187, A194547, A194549, A195822, A208478.

Terms a(1)-a(22) confirmed and additional terms added by John W. Layman, Mar 10 2012

A237751 Number of partitions of n such that 2*(greatest part) < (number of parts).

Original entry on oeis.org

0, 0, 1, 1, 1, 2, 3, 4, 6, 8, 10, 14, 18, 24, 32, 41, 52, 67, 85, 107, 135, 169, 210, 263, 324, 400, 493, 604, 736, 899, 1091, 1322, 1599, 1929, 2319, 2787, 3336, 3989, 4760, 5669, 6734, 7994, 9465, 11192, 13211, 15571, 18319, 21531, 25257, 29594, 34626

Offset: 1

Clark Kimberling, Feb 13 2014

Also, the number of partitions of n such that (greatest part) > 2*(number of parts); hence, the number of partitions of n such that (rank + greatest part) < 0.

			a(6) = 2 counts these partitions:  21111, 111111.

Seiichi Manyama, Table of n, a(n) for n = 1..1000

Cf. A064173, A237752-A237757, A000041.

z = 55; Table[Count[IntegerPartitions[n], p_ /; 2 Max[p] < Length[p]], {n, z}]

a(n) = A000041(n) - A237755(n).

A237752 Number of partitions of n such that 2*(greatest part) <= (number of parts).

Original entry on oeis.org

0, 1, 1, 1, 2, 3, 4, 6, 7, 10, 13, 18, 23, 31, 39, 50, 64, 82, 102, 130, 162, 203, 252, 313, 384, 475, 580, 710, 864, 1053, 1273, 1544, 1859, 2240, 2688, 3224, 3851, 4602, 5476, 6514, 7727, 9160, 10826, 12791, 15072, 17747, 20853, 24481, 28679, 33577, 39231

Offset: 1

Clark Kimberling, Feb 13 2014

Also, the number of partitions of n such that (greatest part) >= 2*(number of parts); hence, the number of partitions of n such that (rank + greatest part) <= 0.

Also, the number of partitions p of n such that max(max(p), 2*(number of parts of p)) is a part of p.

			The partitions of 6 that do not qualify are 22311, 21111, 111111, so that a(6) = 11 - 3 = 8.

Seiichi Manyama, Table of n, a(n) for n = 1..1000

Cf. A064173, A237751, A237753-A237757, A000041.

z = 50; Table[Count[IntegerPartitions[n], p_ /; 2 Max[p] <= Length[p]], {n, z}]
(* also *)
Table[Count[IntegerPartitions[n], p_ /; MemberQ[p, Max[Max[p],2*Length[p]]]], {n, 50}]

a(n) = A000041(n) - A237754(n).

A340827 Number of strict integer partitions of n into divisors of n whose length also divides n.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 5, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 5, 1, 1, 1, 3, 1, 2, 1, 1, 1, 1, 1, 6, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 25, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 18, 1, 1, 1, 1, 1, 2, 1, 3, 1, 1, 1, 17, 1, 1, 1, 1, 1, 14, 1, 1, 1, 1, 1, 12, 1, 1, 1, 3, 1, 2, 1, 1, 1

Offset: 1

Gus Wiseman, Feb 01 2021

nonn

The first element not in A326715 that is however a Heinz number of these partitions is 273.

			The a(n) partitions for n = 6, 12, 24, 90, 84:
  6       12        24            90                      84
  3,2,1   6,4,2     12,8,4        45,30,15                42,28,14
          6,3,2,1   12,6,4,2      45,30,9,5,1             42,21,14,7
                    12,8,3,1      45,18,15,9,3            42,28,12,2
                    8,6,4,3,2,1   45,30,10,3,2            42,28,6,4,3,1
                                  45,18,15,10,2           42,28,7,4,2,1
                                  45,30,6,5,3,1           42,14,12,7,6,3
                                  45,30,9,3,2,1           42,21,12,4,3,2
                                  45,15,10,9,6,5          42,21,12,6,2,1
                                  45,18,10,9,5,3          42,21,14,4,2,1
                                  45,18,10,9,6,2          28,21,14,12,6,3
                                  45,18,15,6,5,1          28,21,14,12,7,2
                                  45,18,15,9,2,1          42,21,7,6,4,3,1
                                  30,18,15,10,6,5,3,2,1   42,14,12,7,4,3,2
                                                          42,14,12,7,6,2,1
                                                          28,21,14,12,4,3,2
                                                          28,21,14,12,6,2,1

David A. Corneth, Table of n, a(n) for n = 1..10000 (first 2519 terms from Antti Karttunen)
Antti Karttunen, Scheme program for computing this sequence

Note: A-numbers of Heinz-number sequences are in parentheses below.
The non-strict case is A326842 (A326847).
A018818 = partitions using divisors (A326841).
A047993 = balanced partitions (A106529).
A067538 = partitions whose length/maximum divides sum (A316413/A326836).
A072233 = partitions by sum and length, with strict case A008289.
A102627 = strict partitions whose length divides sum.
A326850 = strict partitions whose maximum part divides sum.
A326851 = strict partitions w/ length and max dividing sum.
A340828 = strict partitions w/ length divisible by max.
A340829 = strict partitions w/ Heinz number divisible by sum.
A340830 = strict partitions w/ parts divisible by length.
Cf. A064173, A143773 (A316428), A168659, A326843 (A326837), A326852, A330950 (A324851), A340851, A340853.

Table[Length[Select[IntegerPartitions[n,All,Divisors[n]],UnsameQ@@#&&Divisible[n,Length[#]]&]],{n,30}]

A340827(n, divsleft=List(divisors(n)), rest=n, len=0) = if(rest<=0, !rest && !(n%len), my(s=0, d); forstep(i=#divsleft, 1, -1, d = divsleft[i]; listpop(divsleft,i); if(rest>=d, s += A340827(n, divsleft, rest-d, 1+len))); (s)); \\ Antti Karttunen, Feb 22 2023

;; See the Links-section. - Antti Karttunen, Feb 22 2023

Data section extended up to a(105) by Antti Karttunen, Feb 22 2023

A123975 Number of Garden of Eden partitions of n in Bulgarian Solitaire.

Original entry on oeis.org

0, 0, 1, 1, 2, 3, 5, 7, 10, 14, 20, 27, 37, 49, 66, 86, 113, 147, 190, 243, 311, 394, 499, 627, 786, 980, 1220, 1510, 1865, 2294, 2816, 3443, 4202, 5110, 6203, 7507, 9067, 10923, 13135, 15755, 18865, 22540, 26885, 32001, 38032, 45112, 53430, 63171

Offset: 1

Vladeta Jovovic, Nov 23 2006

a(n) gives the number of times n occurs in A225794. - Antti Karttunen, Jul 27 2013

Seiichi Manyama, Table of n, a(n) for n = 1..10000
Brian Hopkins, Michael A. Jones, Shift-Induced Dynamical Systems on Partitions and Compositions, Electron. J. Combin. 13 (2006), Research Paper 80.
Brian Hopkins and James A. Sellers, Exact enumeration of Garden of Eden partitions, INTEGERS: Electronic Journal of Combinatorial Number Theory 7(2) (2007), #A19.

Cf. A000041, A045943, A064173, A101198, A237831.

Cf. A227753, A225794, A226062.

p:=product(1/(1-q^i), i=1..200)*sum((-1)^(r-1)*q^((3*r^2+3*r)/2), r=1..200):s:=series(p, q, 200): for j from 0 to 199 do printf(`%d,`,coeff(s, q, j)) od: # James Sellers, Nov 30 2006

my(N=50, x='x+O('x^N)); concat([0, 0], Vec(1/prod(k=1, N, 1-x^k)*sum(k=1, N, (-1)^(k-1)*x^(3*k*(k+1)/2)))) \\ Seiichi Manyama, May 21 2023

a(n) = A064173(n) - A101198(n).
a(n) = Sum_{j>=1} (-1)^(j+1)*p(n-b(j)) where b(j) = 3*j*(j+1)/2 (A045943) and p(n) is the number of partitions of n (see A000041). See Hopkins & Sellers. - Michel Marcus, Sep 26 2018
a(n) ~ exp(Pi*sqrt(2*n/3)) / (8*n*sqrt(3)) * (1 - (1/(2*Pi) + 19*Pi/144) / sqrt(n/6)). - Vaclav Kotesovec, May 26 2023

More terms from James Sellers, Nov 30 2006

A237754 Number of partitions of n such that 2*(greatest part) > (number of parts).

Original entry on oeis.org

1, 1, 2, 4, 5, 8, 11, 16, 23, 32, 43, 59, 78, 104, 137, 181, 233, 303, 388, 497, 630, 799, 1003, 1262, 1574, 1961, 2430, 3008, 3701, 4551, 5569, 6805, 8284, 10070, 12195, 14753, 17786, 21413, 25709, 30824, 36856, 44014, 52435, 62384, 74062, 87811, 103901

Offset: 1

Clark Kimberling, Feb 13 2014

Also, the number of partitions of n such that (greatest part) < 2*(number of parts); hence, the number of partitions of n such that (rank + greatest part) > 0.

Also, the number of partitions p of n such that max(max(p), 2*(number of parts of p)) is not a part of p.

			a(5) = 5 counts these partitions:  5, 41, 32, 311, 221.

Seiichi Manyama, Table of n, a(n) for n = 1..1000

Cf. A064173, A237751-A237753, A237755-A237757, A000041.

z = 50; Table[Count[IntegerPartitions[n], p_ /; 2 Max[p] > Length[p]], {n, z}]

my(N=66, x='x+O('x^N)); Vec(sum(k=1, N, x^k*prod(j=1, k, (1-x^(2*k+j-2))/(1-x^j)))) \\ Seiichi Manyama, Jan 25 2022

a(n) = A000041(n) - A237752(n).

G.f.: Sum_{k>=1} x^k * Product_{j=1..k} (1-x^(2*k+j-2))/(1-x^j). - Seiichi Manyama, Jan 25 2022

cp's OEIS Frontend

A340605 Heinz numbers of integer partitions of even positive rank.

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A208478 Triangle read by rows: T(n,k) = number of partitions of n with positive k-th rank.

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A340787 Heinz numbers of integer partitions of positive rank.

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A340828 Number of strict integer partitions of n whose maximum part is a multiple of their length.

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A208482 Triangle read by rows: T(n,k) = sum of positive k-th ranks of all partitions of n.

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A237751 Number of partitions of n such that 2*(greatest part) < (number of parts).

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A237752 Number of partitions of n such that 2*(greatest part) <= (number of parts).

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A340827 Number of strict integer partitions of n into divisors of n whose length also divides n.

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