A340787 - OEIS

%I #13 Apr 09 2021 09:41:15

%S 3,5,7,10,11,13,14,15,17,19,21,22,23,25,26,28,29,31,33,34,35,37,38,39,

%T 41,42,43,44,46,47,49,51,52,53,55,57,58,59,61,62,63,65,66,67,68,69,70,

%U 71,73,74,76,77,78,79,82,83,85,86,87,88,89,91,92,93,94,95

%N Heinz numbers of integer partitions of positive rank.

%C 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.

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

%H Freeman J. Dyson, <a href="https://doi.org/10.1016/S0021-9800(69)80006-2">A new symmetry of partitions</a>, Journal of Combinatorial Theory 7.1 (1969): 56-61.

%H FindStat, <a href="http://www.findstat.org/StatisticsDatabase/St000145">St000145: The Dyson rank of a partition</a>

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

%e The sequence of partitions together with their Heinz numbers begins:

%e      3: (2)      28: (4,1,1)    49: (4,4)      69: (9,2)

%e      5: (3)      29: (10)       51: (7,2)      70: (4,3,1)

%e      7: (4)      31: (11)       52: (6,1,1)    71: (20)

%e     10: (3,1)    33: (5,2)      53: (16)       73: (21)

%e     11: (5)      34: (7,1)      55: (5,3)      74: (12,1)

%e     13: (6)      35: (4,3)      57: (8,2)      76: (8,1,1)

%e     14: (4,1)    37: (12)       58: (10,1)     77: (5,4)

%e     15: (3,2)    38: (8,1)      59: (17)       78: (6,2,1)

%e     17: (7)      39: (6,2)      61: (18)       79: (22)

%e     19: (8)      41: (13)       62: (11,1)     82: (13,1)

%e     21: (4,2)    42: (4,2,1)    63: (4,2,2)    83: (23)

%e     22: (5,1)    43: (14)       65: (6,3)      85: (7,3)

%e     23: (9)      44: (5,1,1)    66: (5,2,1)    86: (14,1)

%e     25: (3,3)    46: (9,1)      67: (19)       87: (10,2)

%e     26: (6,1)    47: (15)       68: (7,1,1)    88: (5,1,1,1)

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

%Y Note: A-numbers of Heinz-number sequences are in parentheses below.

%Y These partitions are counted by A064173.

%Y The odd case is A101707 (A340604).

%Y The even case is A101708 (A340605).

%Y The negative version is (A340788).

%Y A001222 counts prime factors.

%Y A061395 selects the maximum prime index.

%Y A072233 counts partitions by sum and length.

%Y A168659 = partitions whose greatest part divides their length (A340609).

%Y A168659 = partitions whose length divides their greatest part (A340610).

%Y A200750 = partitions whose length and maximum are relatively prime.

%Y - Rank -

%Y A047993 counts partitions of rank 0 (A106529).

%Y A063995/A105806 count partitions by Dyson rank.

%Y A064174 counts partitions of nonnegative/nonpositive rank (A324562/A324521).

%Y A101198 counts partitions of rank 1 (A325233).

%Y A257541 gives the rank of the partition with Heinz number n.

%Y A324520 counts partitions with rank equal to least part (A324519).

%Y A340601 counts partitions of even rank (A340602), with strict case A117192.

%Y A340692 counts partitions of odd rank (A340603), with strict case A117193.

%Y Cf. A003114, A006141, A039900, A056239, A096401, A112798, A117409, A316413, A324517, A325134, A326845, A340828.

%K nonn

%O 1,1

%A _Gus Wiseman_, Jan 29 2021