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Heinz numbers of integer partitions with Dyson rank -1.
3

%I #6 Apr 13 2019 22:16:52

%S 4,12,18,27,40,60,90,100,112,135,150,168,225,250,252,280,352,375,378,

%T 392,420,528,567,588,625,630,700,792,832,880,882,945,980,1050,1188,

%U 1232,1248,1320,1323,1372,1470,1575,1750,1782,1848,1872,1936,1980,2058,2080

%N Heinz numbers of integer partitions with Dyson rank -1.

%C Numbers whose maximum prime index is one fewer than their number of prime indices counted with multiplicity.

%C The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).

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

%e The sequence of terms together with their prime indices begins:

%e 4: {1,1}

%e 12: {1,1,2}

%e 18: {1,2,2}

%e 27: {2,2,2}

%e 40: {1,1,1,3}

%e 60: {1,1,2,3}

%e 90: {1,2,2,3}

%e 100: {1,1,3,3}

%e 112: {1,1,1,1,4}

%e 135: {2,2,2,3}

%e 150: {1,2,3,3}

%e 168: {1,1,1,2,4}

%e 225: {2,2,3,3}

%e 250: {1,3,3,3}

%e 252: {1,1,2,2,4}

%e 280: {1,1,1,3,4}

%e 352: {1,1,1,1,1,5}

%e 375: {2,3,3,3}

%e 378: {1,2,2,2,4}

%e 392: {1,1,1,4,4}

%t Select[Range[1000],PrimePi[FactorInteger[#][[-1,1]]]-PrimeOmega[#]==-1&]

%Y Positions of -1's in A257541.

%Y Cf. A001222, A047993, A056239, A061395, A063995, A101198, A106529, A112798, A257990, A263297, A325225, A325233, A325235.

%K nonn

%O 1,1

%A _Gus Wiseman_, Apr 13 2019