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A348550 Heinz numbers of integer partitions whose length is 2/3 their sum, rounded down.

%I A348550 #11 Nov 09 2021 18:42:09
%S A348550 1,3,6,9,10,18,20,36,40,54,56,60,108,112,120,216,224,240,324,336,352,
%T A348550 360,400,648,672,704,720,800,1296,1344,1408,1440,1600,1664,1944,2016,
%U A348550 2112,2160,2240,2400,3328,3888,4032,4224,4320,4480,4800,6656,7776,8064,8448
%N A348550 Heinz numbers of integer partitions whose length is 2/3 their sum, rounded down.
%C A348550 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 A348550 A001222(a(n)) = floor(2*A056239(a(n))/3).
%e A348550 The terms and their prime indices begin:
%e A348550     1: {}
%e A348550     3: {2}
%e A348550     6: {1,2}
%e A348550     9: {2,2}
%e A348550    10: {1,3}
%e A348550    18: {1,2,2}
%e A348550    20: {1,1,3}
%e A348550    36: {1,1,2,2}
%e A348550    40: {1,1,1,3}
%e A348550    54: {1,2,2,2}
%e A348550    56: {1,1,1,4}
%e A348550    60: {1,1,2,3}
%e A348550   108: {1,1,2,2,2}
%e A348550   112: {1,1,1,1,4}
%e A348550   120: {1,1,1,2,3}
%e A348550   216: {1,1,1,2,2,2}
%e A348550   224: {1,1,1,1,1,4}
%e A348550   240: {1,1,1,1,2,3}
%t A348550 Select[Range[1000],Floor[2*Total[Cases[FactorInteger[#],{p_,k_}:>k*PrimePi[p]]]/3]==PrimeOmega[#]&]
%o A348550 (PARI)
%o A348550 A056239(n) = { my(f); if(1==n, 0, f=factor(n); sum(i=1, #f~, f[i,2] * primepi(f[i,1]))); }
%o A348550 isA348550(n) = (bigomega(n)==floor((2/3)*A056239(n))); \\ _Antti Karttunen_, Nov 08 2021
%Y A348550 The partitions with these as Heinz numbers are counted by A108711.
%Y A348550 An adjoint version is A347452, counted by A119620.
%Y A348550 The unrounded version is A348384, counted by A035377.
%Y A348550 A001222 counts prime factors with multiplicity.
%Y A348550 A056239 adds up prime indices, row sums of A112798.
%Y A348550 A316524 gives the alternating sum of prime indices, reverse A344616.
%Y A348550 A344606 counts alternating permutations of prime factors.
%Y A348550 Cf. A001105, A028982, A028260, A119899, A316413, A346703, A346704.
%K A348550 nonn
%O A348550 1,2
%A A348550 _Gus Wiseman_, Nov 05 2021