A340787 Heinz numbers of integer partitions of positive rank.

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

1,1

Comments

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.

Links

Table of n, a(n) for n=1..66.

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

Formula

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

Example

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)

Mathematica

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

Crossrefs

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.

Sequence in context: A189293 A140607 A352487 * A286607 A204827 A366882

Adjacent sequences: A340784 A340785 A340786 * A340788 A340789 A340790

Keyword

nonn

Author

Gus Wiseman, Jan 29 2021

Status

approved