A347452 - OEIS

%I #9 Oct 29 2021 09:06:55

%S 1,2,6,12,36,40,72,80,216,224,240,432,448,480,1296,1344,1408,1440,

%T 1600,2592,2688,2816,2880,3200,6656,7776,8064,8448,8640,8960,9600,

%U 13312,15552,16128,16896,17280,17920,19200,34816,39936,46656,48384,50176,50688,51840

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

%C 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 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 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 Counting the partitions with these Heinz numbers gives A119620 with zeros interspersed every three terms.

%e The initial terms and their prime indices:

%e       1: {}

%e       2: {1}

%e       6: {1,2}

%e      12: {1,1,2}

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

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

%e      72: {1,1,1,2,2}

%e      80: {1,1,1,1,3}

%e     216: {1,1,1,2,2,2}

%e     224: {1,1,1,1,1,4}

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

%e     432: {1,1,1,1,2,2,2}

%e     448: {1,1,1,1,1,1,4}

%e     480: {1,1,1,1,1,2,3}

%e    1296: {1,1,1,1,2,2,2,2}

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

%e    1408: {1,1,1,1,1,1,1,5}

%e    1440: {1,1,1,1,1,2,2,3}

%e    1600: {1,1,1,1,1,1,3,3}

%t Select[Range[1000],Total[Cases[FactorInteger[#],{p_,k_}:>k*PrimePi[p]]]==Floor[3*PrimeOmega[#]/2]&]

%Y Counting terms by Heinz weight (in A032766) gives A119620.

%Y An adjoint version is A348550, counted by A108711.

%Y A000041 counts partitions.

%Y A001222 counts prime factors with multiplicity.

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

%Y A316524 gives the alternating sum of prime indices (reverse: A344616).

%Y A344606 counts wiggly permutations of prime factors.

%Y Cf. A000070, A000097, A028982, A236914, A316413, A347457, A348551.

%K nonn

%O 1,2

%A _Gus Wiseman_, Oct 28 2021