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Numbers whose prime indices have alternating sum -1.
10

%I #39 Jul 15 2021 15:08:11

%S 6,15,24,35,54,60,77,96,135,140,143,150,216,221,240,294,308,315,323,

%T 375,384,437,486,540,560,572,600,667,693,726,735,864,875,884,899,960,

%U 1014,1147,1176,1215,1232,1260,1287,1292,1350,1500,1517,1536,1715,1734,1748

%N Numbers whose prime indices have alternating sum -1.

%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 The alternating sum of a sequence (y_1,...,y_k) is Sum_i (-1)^(i-1) y_i. Of course, the alternating sum of prime indices is also the reverse-alternating sum of reversed prime indices.

%C Also numbers with even Omega (A001222) and exactly one odd conjugate prime index. Conjugate prime indices are listed by A321650, ranked by A122111.

%e The initial terms and their prime indices:

%e 6: {1,2}

%e 15: {2,3}

%e 24: {1,1,1,2}

%e 35: {3,4}

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

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

%e 77: {4,5}

%e 96: {1,1,1,1,1,2}

%e 135: {2,2,2,3}

%e 140: {1,1,3,4}

%e 143: {5,6}

%e 150: {1,2,3,3}

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

%e 221: {6,7}

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

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

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

%t Select[Range[0,100],ats[primeMS[#]]==-1&]

%Y These multisets are counted by A000070.

%Y The k = 0 version is A000290, counted by A000041.

%Y The k = 1 version is A001105.

%Y The k > 0 version is A026424.

%Y These are the positions of -1's in A316524.

%Y The k = 2 version is A345960.

%Y The k = -2 version is A345962.

%Y A000984/A345909/A345911 count/rank compositions with alternating sum 1.

%Y A001791/A345910/A345912 count/rank compositions with alternating sum -1.

%Y A027187 counts partitions with reverse-alternating sum <= 0.

%Y A056239 adds up prime indices, row sums of A112798.

%Y A097805 counts compositions by alternating (or reverse-alternating) sum.

%Y A103919 counts partitions by sum and alternating sum (reverse: A344612).

%Y A325534/A325535 count separable/inseparable partitions.

%Y A344607 counts partitions with reverse-alternating sum >= 0.

%Y A344616 gives the alternating sum of reversed prime indices.

%Y Cf. A000097, A028260, A035363, A236913, A239830, A341446, A344609, A344610, A344651, A345919, A345961.

%K nonn

%O 1,1

%A _Gus Wiseman_, Jul 12 2021