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Heinz numbers of integer partitions whose product equals their length.
1

%I #9 May 20 2022 23:04:08

%S 2,6,20,36,56,176,240,416,864,1088,1344,2432,3200,5888,8448,14848,

%T 23040,31744,35840,39936,75776,167936,208896,331776,352256,450560,

%U 516096,770048,802816,933888,1736704,2457600,3866624,4259840,4521984,7995392,12976128,17563648

%N Heinz numbers of integer partitions whose product equals their length.

%C 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 The terms together with their prime indices begin:

%e 2: {1}

%e 6: {1,2}

%e 20: {1,1,3}

%e 36: {1,1,2,2}

%e 56: {1,1,1,4}

%e 176: {1,1,1,1,5}

%e 240: {1,1,1,1,2,3}

%e 416: {1,1,1,1,1,6}

%e 864: {1,1,1,1,1,2,2,2}

%e 1088: {1,1,1,1,1,1,7}

%e 1344: {1,1,1,1,1,1,2,4}

%e 2432: {1,1,1,1,1,1,1,8}

%e 3200: {1,1,1,1,1,1,1,3,3}

%e 5888: {1,1,1,1,1,1,1,1,9}

%e 8448: {1,1,1,1,1,1,1,1,2,5}

%e 14848: {1,1,1,1,1,1,1,1,1,10}

%e 23040: {1,1,1,1,1,1,1,1,1,2,2,3}

%e 31744: {1,1,1,1,1,1,1,1,1,1,11}

%e 35840: {1,1,1,1,1,1,1,1,1,1,3,4}

%e 39936: {1,1,1,1,1,1,1,1,1,1,2,6}

%e 75776: {1,1,1,1,1,1,1,1,1,1,1,12}

%t Select[Range[1000],Times@@Cases[If[#==1,{},FactorInteger[#]],{p_,k_}:>PrimePi[p]^k]==PrimeOmega[#]&]

%Y Length is A001222, counted by A008284, distinct A001221.

%Y Product is A003963, counted by A339095, firsts A318871.

%Y A similar sequence is A353503, counted by A353506.

%Y These partitions are counted by A353698.

%Y A005361 gives product of signature, firsts A353500 (sorted A085629).

%Y A056239 adds up prime indices, row sums of A112798 and A296150.

%Y A124010 gives prime signature, sorted A118914.

%Y A181819 gives prime shadow, with an inverse A181821.

%Y A353394 gives product of shadows of prime indices, firsts A353397.

%Y Cf. A000720, A003586, A114640, A182850, A320325, A325131, A325755, A353399, A353507.

%K nonn

%O 1,1

%A _Gus Wiseman_, May 19 2022