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%I A363625 #8 Aug 16 2023 11:00:07
%S A363625 0,1,2,1,3,3,4,2,2,5,5,5,6,7,4,2,7,3,8,8,6,9,9,6,3,11,4,11,10,6,11,3,
%T A363625 8,13,5,3,12,15,10,10,13,9,14,14,7,17,15,8,4,4,12,17,16,5,7,14,14,19,
%U A363625 17,7,18,21,10,3,9,12,19,20,16,7,20,4,21,23,5,23
%N A363625 Reverse-weighted alternating sum of the integer partition with Heinz number n.
%C A363625 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 A363625 We define the reverse-weighted alternating sum of a sequence (y_1,...,y_k) to be Sum_{i=1..k} (-1)^(k-i) * i * y_{k-i+1}.
%e A363625 The partition with Heinz number 600 is (3,3,2,1,1,1), so a(600) = -1*1 + 2*1 - 3*1 + 4*2 - 5*3 + 6*3 = 9.
%t A363625 prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A363625 revaltwtsum[y_]:=Sum[(-1)^(Length[y]-k)*k*y[[-k]],{k,1,Length[y]}];
%t A363625 Table[revaltwtsum[Reverse[prix[n]]],{n,100}]
%Y A363625 The non-alternating version is A304818, reverse A318283.
%Y A363625 The unweighted version is A316524, reverse A344616.
%Y A363625 For multisets instead of partitions we have A363620.
%Y A363625 The triangle for this rank statistic is A363623, reverse A363622.
%Y A363625 The reverse version is A363624, for multisets A363619.
%Y A363625 A055396 gives minimum prime index, maximum A061395.
%Y A363625 A112798 lists prime indices, length A001222, sum A056239.
%Y A363625 A264034 counts partitions by weighted sum, reverse A358194.
%Y A363625 A320387 counts multisets by weighted sum, reverse A007294.
%Y A363625 Cf. A046660, A053632, A106529, A124010, A261079, A359361, A359755, A363532, A363621, A363626.
%K A363625 nonn
%O A363625 1,3
%A A363625 _Gus Wiseman_, Jun 15 2023

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