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%I A345960 #8 Jul 15 2021 15:08:19
%S A345960 3,12,27,30,48,70,75,108,120,147,154,192,243,270,280,286,300,363,432,
%T A345960 442,480,507,588,616,630,646,675,750,768,867,874,972,1080,1083,1120,
%U A345960 1144,1200,1323,1334,1386,1452,1470,1587,1728,1750,1768,1798,1875,1920,2028
%N A345960 Numbers whose prime indices have alternating sum 2.
%C A345960 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 A345960 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 A345960 Also numbers with odd Omega (A001222) and exactly two odd conjugate prime indices. The version for even Omega is A345962, and the union is A345961. Conjugate prime indices are listed by A321650 and ranked by A122111.
%e A345960 The initial terms and their prime indices:
%e A345960     3: {2}
%e A345960    12: {1,1,2}
%e A345960    27: {2,2,2}
%e A345960    30: {1,2,3}
%e A345960    48: {1,1,1,1,2}
%e A345960    70: {1,3,4}
%e A345960    75: {2,3,3}
%e A345960   108: {1,1,2,2,2}
%e A345960   120: {1,1,1,2,3}
%e A345960   147: {2,4,4}
%e A345960   154: {1,4,5}
%e A345960   192: {1,1,1,1,1,1,2}
%e A345960   243: {2,2,2,2,2}
%e A345960   270: {1,2,2,2,3}
%e A345960   280: {1,1,1,3,4}
%e A345960   286: {1,5,6}
%e A345960   300: {1,1,2,3,3}
%t A345960 primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A345960 ats[y_]:=Sum[(-1)^(i-1)*y[[i]],{i,Length[y]}];
%t A345960 Select[Range[0,100],ats[primeMS[#]]==2&]
%Y A345960 These partitions are counted by A000097.
%Y A345960 The k = 0 version is A000290, counted by A000041.
%Y A345960 The k = 1 version is A001105 (reverse: A345958).
%Y A345960 The k > 0 version is A026424.
%Y A345960 These multisets are counted by A120452.
%Y A345960 These are the positions of 2's in A316524 (reverse: A344616).
%Y A345960 The k = -1 version is A345959.
%Y A345960 The version for reversed alternating sum is A345961.
%Y A345960 The k = -2 version is A345962.
%Y A345960 A000984/A345909/A345911 count/rank compositions with alternating sum 1.
%Y A345960 A002054/A345924/A345923 count/rank compositions with alternating sum -2.
%Y A345960 A056239 adds up prime indices, row sums of A112798.
%Y A345960 A088218/A345925/A345922 count/rank compositions with alternating sum 2.
%Y A345960 A097805 counts compositions by alternating (or reverse-alternating) sum.
%Y A345960 A103919 counts partitions by sum and alternating sum (reverse: A344612).
%Y A345960 A325534 and A325535 count separable and inseparable partitions.
%Y A345960 A344606 counts alternating permutations of prime indices.
%Y A345960 Cf. A000037, A000070, A028260, A035363, A239830, A341446, A344609, A344610, A344651, A344741, A345197.
%K A345960 nonn
%O A345960 1,1
%A A345960 _Gus Wiseman_, Jul 12 2021

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