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A345962 Numbers whose prime indices have alternating sum -2.

%I A345962 #7 Jul 15 2021 15:08:37
%S A345962 10,21,40,55,84,90,91,160,187,189,210,220,247,250,336,360,364,391,462,
%T A345962 490,495,525,551,640,713,748,756,810,819,840,858,880,988,1000,1029,
%U A345962 1073,1155,1210,1271,1326,1344,1375,1440,1456,1564,1591,1683,1690,1701,1848
%N A345962 Numbers whose prime indices have alternating sum -2.
%C A345962 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 A345962 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 A345962 Also numbers with even Omega (A001222) and exactly two odd conjugate prime indices. The case of odd Omega is A345960, and the union is A345961.
%e A345962 The initial terms and their prime indices:
%e A345962    10: {1,3}
%e A345962    21: {2,4}
%e A345962    40: {1,1,1,3}
%e A345962    55: {3,5}
%e A345962    84: {1,1,2,4}
%e A345962    90: {1,2,2,3}
%e A345962    91: {4,6}
%e A345962   160: {1,1,1,1,1,3}
%e A345962   187: {5,7}
%e A345962   189: {2,2,2,4}
%e A345962   210: {1,2,3,4}
%e A345962   220: {1,1,3,5}
%e A345962   247: {6,8}
%e A345962   250: {1,3,3,3}
%e A345962   336: {1,1,1,1,2,4}
%e A345962   360: {1,1,1,2,2,3}
%t A345962 primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A345962 ats[y_]:=Sum[(-1)^(i-1)*y[[i]],{i,Length[y]}];
%t A345962 Select[Range[0,100],ats[primeMS[#]]==-2&]
%Y A345962 Below we use k to indicate alternating sum.
%Y A345962 The k = 0 version is A000290, counted by A000041.
%Y A345962 The k = 1 version is A001105 (reverse: A345958).
%Y A345962 The k > 0 version is A026424.
%Y A345962 These are the positions of -2's in A316524.
%Y A345962 These multisets are counted by A344741 (positive 2: A120452).
%Y A345962 The k = -1 version is A345959.
%Y A345962 The k = 2 version is A345960, counted by A000097.
%Y A345962 A002054/A345924/A345923 count/rank compositions with alternating sum -2.
%Y A345962 A056239 adds up prime indices, row sums of A112798.
%Y A345962 A088218/A345925/A345922 count/rank compositions with alternating sum 2.
%Y A345962 A097805 counts compositions by alternating (or reverse-alternating) sum.
%Y A345962 A103919 counts partitions by sum and alternating sum (reverse: A344612).
%Y A345962 A325534 and A325535 count separable and inseparable partitions.
%Y A345962 A344606 counts alternating permutations of prime indices.
%Y A345962 Cf. A000037, A000070, A001791, A027187, A028260, A239830, A341446, A344609, A344616, A344651, A345197, A345910, A345912, A345961.
%K A345962 nonn
%O A345962 1,1
%A A345962 _Gus Wiseman_, Jul 12 2021