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A347452 Heinz numbers of integer partitions whose sum is 3/2 their length, rounded down.

%I A347452 #9 Oct 29 2021 09:06:55
%S A347452 1,2,6,12,36,40,72,80,216,224,240,432,448,480,1296,1344,1408,1440,
%T A347452 1600,2592,2688,2816,2880,3200,6656,7776,8064,8448,8640,8960,9600,
%U A347452 13312,15552,16128,16896,17280,17920,19200,34816,39936,46656,48384,50176,50688,51840
%N A347452 Heinz numbers of integer partitions whose sum is 3/2 their length, rounded down.
%C A347452 Also numbers whose sum of prime indices is 3/2 their number, rounded down, where a prime index of n is a number m such that prime(m) divides n.
%C A347452 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.
%C A347452 The sequence contains n iff A056239(n) = floor(3*A001222(n)/2). Here, A056239 adds up prime indices, and A001222 counts them with multiplicity.
%C A347452 Counting the partitions with these Heinz numbers gives A119620 with zeros interspersed every three terms.
%e A347452 The initial terms and their prime indices:
%e A347452       1: {}
%e A347452       2: {1}
%e A347452       6: {1,2}
%e A347452      12: {1,1,2}
%e A347452      36: {1,1,2,2}
%e A347452      40: {1,1,1,3}
%e A347452      72: {1,1,1,2,2}
%e A347452      80: {1,1,1,1,3}
%e A347452     216: {1,1,1,2,2,2}
%e A347452     224: {1,1,1,1,1,4}
%e A347452     240: {1,1,1,1,2,3}
%e A347452     432: {1,1,1,1,2,2,2}
%e A347452     448: {1,1,1,1,1,1,4}
%e A347452     480: {1,1,1,1,1,2,3}
%e A347452    1296: {1,1,1,1,2,2,2,2}
%e A347452    1344: {1,1,1,1,1,1,2,4}
%e A347452    1408: {1,1,1,1,1,1,1,5}
%e A347452    1440: {1,1,1,1,1,2,2,3}
%e A347452    1600: {1,1,1,1,1,1,3,3}
%t A347452 Select[Range[1000],Total[Cases[FactorInteger[#],{p_,k_}:>k*PrimePi[p]]]==Floor[3*PrimeOmega[#]/2]&]
%Y A347452 Counting terms by Heinz weight (in A032766) gives A119620.
%Y A347452 An adjoint version is A348550, counted by A108711.
%Y A347452 A000041 counts partitions.
%Y A347452 A001222 counts prime factors with multiplicity.
%Y A347452 A056239 adds up prime indices, row sums of A112798.
%Y A347452 A316524 gives the alternating sum of prime indices (reverse: A344616).
%Y A347452 A344606 counts wiggly permutations of prime factors.
%Y A347452 Cf. A000070, A000097, A028982, A236914, A316413, A347457, A348551.
%K A347452 nonn
%O A347452 1,2
%A A347452 _Gus Wiseman_, Oct 28 2021