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%I #7 Jan 30 2021 22:51:47
%S 4,12,16,18,27,40,48,60,64,72,90,100,108,112,135,150,160,162,168,192,
%T 225,240,243,250,252,256,280,288,352,360,375,378,392,400,420,432,448,
%U 528,540,567,588,600,625,630,640,648,672,700,768,792,810,832,880,882
%N Heinz numbers of integer partitions of odd 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>
%F For all terms, A061395(a(n)) - A001222(a(n)) is odd and negative.
%e The sequence of partitions together with their Heinz numbers begins:
%e 4: (1,1) 150: (3,3,2,1)
%e 12: (2,1,1) 160: (3,1,1,1,1,1)
%e 16: (1,1,1,1) 162: (2,2,2,2,1)
%e 18: (2,2,1) 168: (4,2,1,1,1)
%e 27: (2,2,2) 192: (2,1,1,1,1,1,1)
%e 40: (3,1,1,1) 225: (3,3,2,2)
%e 48: (2,1,1,1,1) 240: (3,2,1,1,1,1)
%e 60: (3,2,1,1) 243: (2,2,2,2,2)
%e 64: (1,1,1,1,1,1) 250: (3,3,3,1)
%e 72: (2,2,1,1,1) 252: (4,2,2,1,1)
%e 90: (3,2,2,1) 256: (1,1,1,1,1,1,1,1)
%e 100: (3,3,1,1) 280: (4,3,1,1,1)
%e 108: (2,2,2,1,1) 288: (2,2,1,1,1,1,1)
%e 112: (4,1,1,1,1) 352: (5,1,1,1,1,1)
%e 135: (3,2,2,2) 360: (3,2,2,1,1,1)
%t rk[n_]:=PrimePi[FactorInteger[n][[-1,1]]]-PrimeOmega[n];
%t Select[Range[2,100],OddQ[rk[#]]&&rk[#]<0&]
%Y Note: A-numbers of Heinz-number sequences are in parentheses below.
%Y These partitions are counted by A101707.
%Y The positive version is A101707 (A340604).
%Y The even version is A101708 (A340930).
%Y The not necessarily odd version is A064173 (A340788).
%Y A001222 counts prime factors.
%Y A027193 counts partitions of odd length (A026424).
%Y A047993 counts balanced partitions (A106529).
%Y A058695 counts partitions of odd numbers (A300063).
%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 A324516 counts partitions with rank equal to maximum minus minimum part (A324515).
%Y A324518 counts partitions with rank equal to greatest part (A324517).
%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, A056239, A096401, A117193, A117409, A325134, A326845, A340604, A340605, A340787, A340854/A340855.
%K nonn
%O 1,1
%A _Gus Wiseman_, Jan 29 2021
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