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A348384 Heinz numbers of integer partitions whose length is 2/3 their sum.

%I A348384 #12 Dec 17 2021 20:29:03
%S A348384 1,6,36,40,216,224,240,1296,1344,1408,1440,1600,6656,7776,8064,8448,
%T A348384 8640,8960,9600,34816,39936,46656,48384,50176,50688,51840,53760,56320,
%U A348384 57600,64000,155648,208896,239616,266240,279936,290304,301056,304128,311040,315392
%N A348384 Heinz numbers of integer partitions whose length is 2/3 their sum.
%C A348384 The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k), so these are numbers whose sum of prime indices is 3/2 their number. Counting the partitions with these Heinz numbers gives A035377(n) = A000041(n/3) if n is a multiple of 3, otherwise 0.
%H A348384 David A. Corneth, <a href="/A348384/b348384.txt">Table of n, a(n) for n = 1..10000</a>
%F A348384 The sequence contains n iff A056239(n) = 3*A001222(n)/2. Here, A056239 adds up prime indices, while A001222 counts them with multiplicity.
%F A348384 Intersection of A028260 and A347452.
%e A348384 The terms and their prime indices begin:
%e A348384      1: {}
%e A348384      6: {1,2}
%e A348384     36: {1,1,2,2}
%e A348384     40: {1,1,1,3}
%e A348384    216: {1,1,1,2,2,2}
%e A348384    224: {1,1,1,1,1,4}
%e A348384    240: {1,1,1,1,2,3}
%e A348384   1296: {1,1,1,1,2,2,2,2}
%e A348384   1344: {1,1,1,1,1,1,2,4}
%e A348384   1408: {1,1,1,1,1,1,1,5}
%e A348384   1440: {1,1,1,1,1,2,2,3}
%e A348384   1600: {1,1,1,1,1,1,3,3}
%e A348384   6656: {1,1,1,1,1,1,1,1,1,6}
%e A348384   7776: {1,1,1,1,1,2,2,2,2,2}
%t A348384 Select[Range[1000],2*Total[Cases[FactorInteger[#],{p_,k_}:>k*PrimePi[p]]]==3*PrimeOmega[#]&]
%o A348384 (PARI)
%o A348384 A056239(n) = { my(f); if(1==n, 0, f=factor(n); sum(i=1, #f~, f[i, 2] * primepi(f[i, 1]))); }
%o A348384 isA348384(n) = (A056239(n)==(3/2)*bigomega(n)); \\ _Antti Karttunen_, Nov 22 2021
%Y A348384 These partitions are counted by A035377.
%Y A348384 Rounding down gives A348550 or A347452, counted by A108711 or A119620.
%Y A348384 A000041 counts integer partitions.
%Y A348384 A001222 counts prime factors with multiplicity.
%Y A348384 A056239 adds up prime indices, row sums of A112798.
%Y A348384 A316524 gives the alternating sum of prime indices (reverse: A344616).
%Y A348384 A344606 counts alternating permutations of prime factors.
%Y A348384 Cf. A000070, A000097, A028260, A028982, A032766, A236914, A316413, A347457, A348551.
%K A348384 nonn
%O A348384 1,2
%A A348384 _Gus Wiseman_, Nov 13 2021