%I #11 Apr 08 2021 03:24:33
%S 1,2,5,6,8,9,11,14,17,20,21,23,24,26,30,31,32,35,36,38,39,41,44,45,47,
%T 49,50,54,56,57,58,59,65,66,67,68,73,74,75,80,81,83,84,86,87,91,92,95,
%U 96,97,99,102,103,104,106,109,110,111,120,122,124,125,126,127
%N Heinz numbers of integer partitions of even rank.
%C The Dyson rank of a nonempty partition is its maximum part minus its length. The rank of an empty partition is 0.
%C 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.
%H FindStat, <a href="http://www.findstat.org/StatisticsDatabase/St000145">St000145: The Dyson rank of a partition</a>
%F Either n = 1 or A061395(n) - A001222(n) is even.
%e The sequence of partitions with their Heinz numbers begins:
%e 1: () 31: (11) 58: (10,1)
%e 2: (1) 32: (1,1,1,1,1) 59: (17)
%e 5: (3) 35: (4,3) 65: (6,3)
%e 6: (2,1) 36: (2,2,1,1) 66: (5,2,1)
%e 8: (1,1,1) 38: (8,1) 67: (19)
%e 9: (2,2) 39: (6,2) 68: (7,1,1)
%e 11: (5) 41: (13) 73: (21)
%e 14: (4,1) 44: (5,1,1) 74: (12,1)
%e 17: (7) 45: (3,2,2) 75: (3,3,2)
%e 20: (3,1,1) 47: (15) 80: (3,1,1,1,1)
%e 21: (4,2) 49: (4,4) 81: (2,2,2,2)
%e 23: (9) 50: (3,3,1) 83: (23)
%e 24: (2,1,1,1) 54: (2,2,2,1) 84: (4,2,1,1)
%e 26: (6,1) 56: (4,1,1,1) 86: (14,1)
%e 30: (3,2,1) 57: (8,2) 87: (10,2)
%t Select[Range[100],EvenQ[PrimePi[FactorInteger[#][[-1,1]]]-PrimeOmega[#]]&]
%Y Taking only length gives A001222.
%Y Taking only maximum part gives A061395.
%Y These partitions are counted by A340601.
%Y The complement is A340603.
%Y The case of positive rank is A340605.
%Y - Rank -
%Y A047993 counts partitions of rank 0 (A106529).
%Y A101198 counts partitions of rank 1 (A325233).
%Y A101707 counts partitions of odd positive rank (A340604).
%Y A101708 counts partitions of even positive rank (A340605).
%Y A257541 gives the rank of the partition with Heinz number n.
%Y A324516 counts partitions with rank = maximum minus minimum part (A324515).
%Y A340653 counts factorizations of rank 0.
%Y A340692 counts partitions of odd rank (A340603).
%Y - Even -
%Y A024430 counts set partitions of even length.
%Y A027187 counts partitions of even length (A028260).
%Y A027187 (also) counts partitions of even maximum (A244990).
%Y A034008 counts compositions of even length.
%Y A035363 counts partitions into even parts (A066207).
%Y A052841 counts ordered set partitions of even length.
%Y A058696 counts partitions of even numbers (A300061).
%Y A067661 counts strict partitions of even length (A030229).
%Y A236913 counts even-length partitions of even numbers (A340784).
%Y A339846 counts factorizations of even length.
%Y Cf. A000041, A006141, A056239, A072233, A112798, A168659, A325134, A326836, A326845, A340386, A340387.
%K nonn
%O 1,2
%A _Gus Wiseman_, Jan 21 2021