A348550 - OEIS

%I #11 Nov 09 2021 18:42:09

%S 1,3,6,9,10,18,20,36,40,54,56,60,108,112,120,216,224,240,324,336,352,

%T 360,400,648,672,704,720,800,1296,1344,1408,1440,1600,1664,1944,2016,

%U 2112,2160,2240,2400,3328,3888,4032,4224,4320,4480,4800,6656,7776,8064,8448

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

%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.

%F A001222(a(n)) = floor(2*A056239(a(n))/3).

%e The terms and their prime indices begin:

%e     1: {}

%e     3: {2}

%e     6: {1,2}

%e     9: {2,2}

%e    10: {1,3}

%e    18: {1,2,2}

%e    20: {1,1,3}

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

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

%e    54: {1,2,2,2}

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

%e    60: {1,1,2,3}

%e   108: {1,1,2,2,2}

%e   112: {1,1,1,1,4}

%e   120: {1,1,1,2,3}

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

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

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

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

%o (PARI)

%o A056239(n) = { my(f); if(1==n, 0, f=factor(n); sum(i=1, #f~, f[i,2] * primepi(f[i,1]))); }

%o isA348550(n) = (bigomega(n)==floor((2/3)*A056239(n))); \\ _Antti Karttunen_, Nov 08 2021

%Y The partitions with these as Heinz numbers are counted by A108711.

%Y An adjoint version is A347452, counted by A119620.

%Y The unrounded version is A348384, counted by A035377.

%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 alternating permutations of prime factors.

%Y Cf. A001105, A028982, A028260, A119899, A316413, A346703, A346704.

%K nonn

%O 1,2

%A _Gus Wiseman_, Nov 05 2021