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A353699 Heinz numbers of integer partitions whose product equals their length.

%I A353699 #9 May 20 2022 23:04:08
%S A353699 2,6,20,36,56,176,240,416,864,1088,1344,2432,3200,5888,8448,14848,
%T A353699 23040,31744,35840,39936,75776,167936,208896,331776,352256,450560,
%U A353699 516096,770048,802816,933888,1736704,2457600,3866624,4259840,4521984,7995392,12976128,17563648
%N A353699 Heinz numbers of integer partitions whose product equals their length.
%C A353699 The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). This gives a bijective correspondence between positive integers and integer partitions.
%e A353699 The terms together with their prime indices begin:
%e A353699       2: {1}
%e A353699       6: {1,2}
%e A353699      20: {1,1,3}
%e A353699      36: {1,1,2,2}
%e A353699      56: {1,1,1,4}
%e A353699     176: {1,1,1,1,5}
%e A353699     240: {1,1,1,1,2,3}
%e A353699     416: {1,1,1,1,1,6}
%e A353699     864: {1,1,1,1,1,2,2,2}
%e A353699    1088: {1,1,1,1,1,1,7}
%e A353699    1344: {1,1,1,1,1,1,2,4}
%e A353699    2432: {1,1,1,1,1,1,1,8}
%e A353699    3200: {1,1,1,1,1,1,1,3,3}
%e A353699    5888: {1,1,1,1,1,1,1,1,9}
%e A353699    8448: {1,1,1,1,1,1,1,1,2,5}
%e A353699   14848: {1,1,1,1,1,1,1,1,1,10}
%e A353699   23040: {1,1,1,1,1,1,1,1,1,2,2,3}
%e A353699   31744: {1,1,1,1,1,1,1,1,1,1,11}
%e A353699   35840: {1,1,1,1,1,1,1,1,1,1,3,4}
%e A353699   39936: {1,1,1,1,1,1,1,1,1,1,2,6}
%e A353699   75776: {1,1,1,1,1,1,1,1,1,1,1,12}
%t A353699 Select[Range[1000],Times@@Cases[If[#==1,{},FactorInteger[#]],{p_,k_}:>PrimePi[p]^k]==PrimeOmega[#]&]
%Y A353699 Length is A001222, counted by A008284, distinct A001221.
%Y A353699 Product is A003963, counted by A339095, firsts A318871.
%Y A353699 A similar sequence is A353503, counted by A353506.
%Y A353699 These partitions are counted by A353698.
%Y A353699 A005361 gives product of signature, firsts A353500 (sorted A085629).
%Y A353699 A056239 adds up prime indices, row sums of A112798 and A296150.
%Y A353699 A124010 gives prime signature, sorted A118914.
%Y A353699 A181819 gives prime shadow, with an inverse A181821.
%Y A353699 A353394 gives product of shadows of prime indices, firsts A353397.
%Y A353699 Cf. A000720, A003586, A114640, A182850, A320325, A325131, A325755, A353399, A353507.
%K A353699 nonn
%O A353699 1,1
%A A353699 _Gus Wiseman_, May 19 2022