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A325040 Heinz numbers of integer partitions with the same product of parts as their conjugate.

%I A325040 #5 Mar 26 2019 21:06:22
%S A325040 1,2,6,9,20,30,49,56,70,75,81,84,90,125,176,182,210,264,350,416,441,
%T A325040 532,540,546,624,660,735,910,1088,1100,1260,1378,1386,1443,1520,1560,
%U A325040 1624,1632,1715,2100,2310,2401,2405,2432,2489,2600,3024,3267,3276,3648,3744
%N A325040 Heinz numbers of integer partitions with the same product of parts as their conjugate.
%C A325040 For example, 182 is the Heinz number of (6,4,1) with product 24 and conjugate (3,2,2,2,1,1) with product also 24.
%C A325040 The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1) * ... * prime(y_k).
%C A325040 The enumeration of these partitions by sum is given by A325039.
%e A325040 The sequence of terms together with their prime indices begins:
%e A325040     1: {}
%e A325040     2: {1}
%e A325040     6: {1,2}
%e A325040     9: {2,2}
%e A325040    20: {1,1,3}
%e A325040    30: {1,2,3}
%e A325040    49: {4,4}
%e A325040    56: {1,1,1,4}
%e A325040    70: {1,3,4}
%e A325040    75: {2,3,3}
%e A325040    81: {2,2,2,2}
%e A325040    84: {1,1,2,4}
%e A325040    90: {1,2,2,3}
%e A325040   125: {3,3,3}
%e A325040   176: {1,1,1,1,5}
%e A325040   182: {1,4,6}
%e A325040   210: {1,2,3,4}
%e A325040   264: {1,1,1,2,5}
%e A325040   350: {1,3,3,4}
%e A325040   416: {1,1,1,1,1,6}
%t A325040 priptn[n_]:=If[n==1,{},Reverse[Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]]];
%t A325040 conj[y_]:=If[Length[y]==0,y,Table[Length[Select[y,#>=k&]],{k,1,Max[y]}]];
%t A325040 Select[Range[100],Times@@priptn[#]==Times@@conj[priptn[#]]&]
%Y A325040 Cf. A000720, A001055, A001222, A003963, A056239, A112798, A122111, A321650.
%Y A325040 Cf. A325039, A325045.
%K A325040 nonn
%O A325040 1,2
%A A325040 _Gus Wiseman_, Mar 25 2019