%I #10 Jun 15 2023 07:32:55

%S 1,6,21,40,50,54,65,132,133,154,210,224,319,340,351,360,374,392,450,

%T 481,486,507,546,598,624,644,731,825,855,969,1007,1029,1054,1144,1210,

%U 1254,1320,1364,1386,1403,1408,1520,1558,1653,1750,1785,1827,1836,1890,1960

%N Positive integers whose prime indices have reverse-weighted alternating sum 0.

%C A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.

%C 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 The prime indices of 360 are {1,1,1,2,2,3}, with reverse-weighted alternating sum 1*3 - 2*2 + 3*2 - 4*1 + 5*1 - 6*1 = 0, so 360 is in the sequence.

%e The terms together with their prime indices begin:

%e 1: {}

%e 6: {1,2}

%e 21: {2,4}

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

%e 50: {1,3,3}

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

%e 65: {3,6}

%e 132: {1,1,2,5}

%e 133: {4,8}

%e 154: {1,4,5}

%e 210: {1,2,3,4}

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

%e 319: {5,10}

%e 340: {1,1,3,7}

%e 351: {2,2,2,6}

%e 360: {1,1,1,2,2,3}

%t prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];

%t revaltwtsum[y_]:=Sum[(-1)^(Length[y]-k)*k*y[[-k]],{k,1,Length[y]}];

%t Select[Range[1000],revaltwtsum[prix[#]]==0&]

%Y The unweighted version is A000290.

%Y Partitions of this type are counted by A363532.

%Y Positions of zeros in A363620 and A363624, reverse A363619 and A363625.

%Y Compositions of this type are counted by A363626.

%Y A053632 counts compositions by weighted sum.

%Y A055396 gives minimum prime index, maximum A061395.

%Y A112798 lists prime indices, length A001222, sum A056239.

%Y A264034 counts partitions by weighted sum, reverse A358194.

%Y A304818 gives weighted sum of prime indices.

%Y A318283 gives weighted sum of reversed prime indices.

%Y A320387 counts multisets by weighted sum.

%Y A344616 gives reverse-alternating sum of prime indices.

%Y A363623 counts partitions by reverse-weighted alternating sum.

%Y Cf. A046660, A106529, A124010, A181819, A261079, A316524, A359674, A363622.

%K nonn

%O 1,2

%A _Gus Wiseman_, Jun 13 2023