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%I #12 Apr 09 2021 09:41:29
%S 8,24,32,36,54,80,81,96,120,128,144,180,200,216,224,270,300,320,324,
%T 336,384,405,450,480,486,500,504,512,560,576,675,704,720,729,750,756,
%U 784,800,840,864,896,1056,1080,1125,1134,1176,1200,1250,1260,1280,1296,1344
%N Heinz numbers of integer partitions of even negative 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>
%e The sequence of partitions together with their Heinz numbers begins:
%e 8: (1,1,1) 270: (3,2,2,2,1)
%e 24: (2,1,1,1) 300: (3,3,2,1,1)
%e 32: (1,1,1,1,1) 320: (3,1,1,1,1,1,1)
%e 36: (2,2,1,1) 324: (2,2,2,2,1,1)
%e 54: (2,2,2,1) 336: (4,2,1,1,1,1)
%e 80: (3,1,1,1,1) 384: (2,1,1,1,1,1,1,1)
%e 81: (2,2,2,2) 405: (3,2,2,2,2)
%e 96: (2,1,1,1,1,1) 450: (3,3,2,2,1)
%e 120: (3,2,1,1,1) 480: (3,2,1,1,1,1,1)
%e 128: (1,1,1,1,1,1,1) 486: (2,2,2,2,2,1)
%e 144: (2,2,1,1,1,1) 500: (3,3,3,1,1)
%e 180: (3,2,2,1,1) 504: (4,2,2,1,1,1)
%e 200: (3,3,1,1,1) 512: (1,1,1,1,1,1,1,1,1)
%e 216: (2,2,2,1,1,1) 560: (4,3,1,1,1,1)
%e 224: (4,1,1,1,1,1) 576: (2,2,1,1,1,1,1,1)
%t rk[n_]:=PrimePi[FactorInteger[n][[-1,1]]]-PrimeOmega[n];
%t Select[Range[2,100],EvenQ[rk[#]]&&rk[#]<0&]
%Y Note: A-numbers of Heinz-number sequences are in parentheses below.
%Y These partitions are counted by A101708.
%Y The positive version is (A340605).
%Y The odd version is A101707 (A340929).
%Y The not necessarily even version is A064173 (A340788).
%Y A001222 counts prime factors.
%Y A027187 counts partitions of even length.
%Y A047993 counts balanced partitions (A106529).
%Y A056239 adds up prime indices.
%Y A058696 counts partitions of even numbers.
%Y A061395 selects the maximum prime index.
%Y A063995/A105806 count partitions by Dyson rank.
%Y A072233 counts partitions by sum and length.
%Y A112798 lists the prime indices of each positive integer.
%Y A168659 counts partitions whose length is divisible by maximum.
%Y A200750 counts partitions whose length and maximum are relatively prime.
%Y - 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).
%Y A340692 counts partitions of odd rank (A340603).
%Y Cf. A003114, A006141, A039900, A117193, A117409, A316413, A324517, A325134, A326845, A340604, A340787, A340828, A340830.
%K nonn
%O 1,1
%A _Gus Wiseman_, Jan 30 2021
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