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A340788 Heinz numbers of integer partitions of negative rank.

%I A340788 #9 Jan 30 2021 22:51:40
%S A340788 4,8,12,16,18,24,27,32,36,40,48,54,60,64,72,80,81,90,96,100,108,112,
%T A340788 120,128,135,144,150,160,162,168,180,192,200,216,224,225,240,243,250,
%U A340788 252,256,270,280,288,300,320,324,336,352,360,375,378,384,392,400,405
%N A340788 Heinz numbers of integer partitions of negative rank.
%C A340788 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 A340788 The Dyson rank of a nonempty partition is its maximum part minus its length. The rank of an empty partition is undefined.
%H A340788 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 A340788 FindStat, <a href="http://www.findstat.org/StatisticsDatabase/St000145">St000145: The Dyson rank of a partition</a>
%F A340788 For all terms A061395(a(n)) < A001222(a(n)).
%e A340788 The sequence of partitions together with their Heinz numbers begins:
%e A340788       4: (1,1)             80: (3,1,1,1,1)
%e A340788       8: (1,1,1)           81: (2,2,2,2)
%e A340788      12: (2,1,1)           90: (3,2,2,1)
%e A340788      16: (1,1,1,1)         96: (2,1,1,1,1,1)
%e A340788      18: (2,2,1)          100: (3,3,1,1)
%e A340788      24: (2,1,1,1)        108: (2,2,2,1,1)
%e A340788      27: (2,2,2)          112: (4,1,1,1,1)
%e A340788      32: (1,1,1,1,1)      120: (3,2,1,1,1)
%e A340788      36: (2,2,1,1)        128: (1,1,1,1,1,1,1)
%e A340788      40: (3,1,1,1)        135: (3,2,2,2)
%e A340788      48: (2,1,1,1,1)      144: (2,2,1,1,1,1)
%e A340788      54: (2,2,2,1)        150: (3,3,2,1)
%e A340788      60: (3,2,1,1)        160: (3,1,1,1,1,1)
%e A340788      64: (1,1,1,1,1,1)    162: (2,2,2,2,1)
%e A340788      72: (2,2,1,1,1)      168: (4,2,1,1,1)
%t A340788 Select[Range[2,100],PrimePi[FactorInteger[#][[-1,1]]]<PrimeOmega[#]&]
%Y A340788 Note: A-numbers of Heinz-number sequences are in parentheses below.
%Y A340788 These partitions are counted by A064173.
%Y A340788 The odd case is A101707 is (A340929).
%Y A340788 The even case is A101708 is (A340930).
%Y A340788 The positive version is (A340787).
%Y A340788 A001222 counts prime factors.
%Y A340788 A061395 selects the maximum prime index.
%Y A340788 A072233 counts partitions by sum and length.
%Y A340788 A168659 counts partitions whose length is divisible by maximum.
%Y A340788 A200750 counts partitions whose length and maximum are relatively prime.
%Y A340788 - Rank -
%Y A340788 A047993 counts partitions of rank 0 (A106529).
%Y A340788 A063995/A105806 count partitions by Dyson rank.
%Y A340788 A064174 counts partitions of nonnegative/nonpositive rank (A324562/A324521).
%Y A340788 A101198 counts partitions of rank 1 (A325233).
%Y A340788 A257541 gives the rank of the partition with Heinz number n.
%Y A340788 A324518 counts partitions with rank equal to greatest part (A324517).
%Y A340788 A324520 counts partitions with rank equal to least part (A324519).
%Y A340788 A340601 counts partitions of even rank (A340602), with strict case A117192.
%Y A340788 A340692 counts partitions of odd rank (A340603), with strict case A117193.
%Y A340788 Cf. A003114, A006141, A039900, A056239, A096401, A112798, A117409, A316413, A325134, A326845, A340604, A340605, A340828.
%K A340788 nonn
%O A340788 1,1
%A A340788 _Gus Wiseman_, Jan 29 2021