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A345959 Numbers whose prime indices have alternating sum -1.

%I A345959 #39 Jul 15 2021 15:08:11
%S A345959 6,15,24,35,54,60,77,96,135,140,143,150,216,221,240,294,308,315,323,
%T A345959 375,384,437,486,540,560,572,600,667,693,726,735,864,875,884,899,960,
%U A345959 1014,1147,1176,1215,1232,1260,1287,1292,1350,1500,1517,1536,1715,1734,1748
%N A345959 Numbers whose prime indices have alternating sum -1.
%C A345959 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 A345959 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 A345959 Also numbers with even Omega (A001222) and exactly one odd conjugate prime index. Conjugate prime indices are listed by A321650, ranked by A122111.
%e A345959 The initial terms and their prime indices:
%e A345959     6: {1,2}
%e A345959    15: {2,3}
%e A345959    24: {1,1,1,2}
%e A345959    35: {3,4}
%e A345959    54: {1,2,2,2}
%e A345959    60: {1,1,2,3}
%e A345959    77: {4,5}
%e A345959    96: {1,1,1,1,1,2}
%e A345959   135: {2,2,2,3}
%e A345959   140: {1,1,3,4}
%e A345959   143: {5,6}
%e A345959   150: {1,2,3,3}
%e A345959   216: {1,1,1,2,2,2}
%e A345959   221: {6,7}
%e A345959   240: {1,1,1,1,2,3}
%t A345959 primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A345959 ats[y_]:=Sum[(-1)^(i-1)*y[[i]],{i,Length[y]}];
%t A345959 Select[Range[0,100],ats[primeMS[#]]==-1&]
%Y A345959 These multisets are counted by A000070.
%Y A345959 The k = 0 version is A000290, counted by A000041.
%Y A345959 The k = 1 version is A001105.
%Y A345959 The k > 0 version is A026424.
%Y A345959 These are the positions of -1's in A316524.
%Y A345959 The k = 2 version is A345960.
%Y A345959 The k = -2 version is A345962.
%Y A345959 A000984/A345909/A345911 count/rank compositions with alternating sum 1.
%Y A345959 A001791/A345910/A345912 count/rank compositions with alternating sum -1.
%Y A345959 A027187 counts partitions with reverse-alternating sum <= 0.
%Y A345959 A056239 adds up prime indices, row sums of A112798.
%Y A345959 A097805 counts compositions by alternating (or reverse-alternating) sum.
%Y A345959 A103919 counts partitions by sum and alternating sum (reverse: A344612).
%Y A345959 A325534/A325535 count separable/inseparable partitions.
%Y A345959 A344607 counts partitions with reverse-alternating sum >= 0.
%Y A345959 A344616 gives the alternating sum of reversed prime indices.
%Y A345959 Cf. A000097, A028260, A035363, A236913, A239830, A341446, A344609, A344610, A344651, A345919, A345961.
%K A345959 nonn
%O A345959 1,1
%A A345959 _Gus Wiseman_, Jul 12 2021