cp's OEIS Frontend

A345958 Numbers whose prime indices have reverse-alternating sum 1.

%I A345958 #8 Jul 15 2021 15:08:01
%S A345958 2,6,8,15,18,24,32,35,50,54,60,72,77,96,98,128,135,140,143,150,162,
%T A345958 200,216,221,240,242,288,294,308,315,323,338,375,384,392,437,450,486,
%U A345958 512,540,560,572,578,600,648,667,693,722,726,735,800,864,875,882,884,899
%N A345958 Numbers whose prime indices have reverse-alternating sum 1.
%C A345958 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 A345958 The reverse-alternating sum of a sequence (y_1,...,y_k) is Sum_i (-1)^(k-i) y_i. Of course, the reverse-alternating sum of prime indices is also the alternating sum of reversed prime indices.
%C A345958 Also numbers with exactly one odd conjugate prime index. Conjugate prime indices are listed by A321650, ranked by A122111.
%e A345958 The initial terms and their prime indices:
%e A345958    2: {1}
%e A345958    6: {1,2}
%e A345958    8: {1,1,1}
%e A345958   15: {2,3}
%e A345958   18: {1,2,2}
%e A345958   24: {1,1,1,2}
%e A345958   32: {1,1,1,1,1}
%e A345958   35: {3,4}
%e A345958   50: {1,3,3}
%e A345958   54: {1,2,2,2}
%e A345958   60: {1,1,2,3}
%e A345958   72: {1,1,1,2,2}
%e A345958   77: {4,5}
%e A345958   96: {1,1,1,1,1,2}
%e A345958   98: {1,4,4}
%t A345958 primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A345958 sats[y_]:=Sum[(-1)^(i-Length[y])*y[[i]],{i,Length[y]}];
%t A345958 Select[Range[100],sats[primeMS[#]]==1&]
%Y A345958 The k > 0 version is A000037.
%Y A345958 These multisets are counted by A000070.
%Y A345958 The k = 0 version is A000290, counted by A000041.
%Y A345958 The version for unreversed-alternating sum is A001105.
%Y A345958 These partitions are counted by A035363.
%Y A345958 These are the positions of 1's in A344616.
%Y A345958 The k = 2 version is A345961, counted by A120452.
%Y A345958 A000984/A345909/A345911 count/rank compositions with alternating sum 1.
%Y A345958 A001791/A345910/A345912 count/rank compositions with alternating sum -1.
%Y A345958 A088218 counts compositions with alternating sum 0, ranked by A344619.
%Y A345958 A025047 counts wiggly compositions.
%Y A345958 A027187 counts partitions with reverse-alternating sum <= 0.
%Y A345958 A056239 adds up prime indices, row sums of A112798.
%Y A345958 A097805 counts compositions by alternating (or reverse-alternating) sum.
%Y A345958 A103919 counts partitions by sum and alternating sum (reverse: A344612).
%Y A345958 A316524 gives the alternating sum of prime indices.
%Y A345958 A325534 and A325535 count separable and inseparable partitions.
%Y A345958 A344606 counts alternating permutations of prime indices.
%Y A345958 A344607 counts partitions with reverse-alternating sum >= 0.
%Y A345958 A344610 counts partitions by sum and positive reverse-alternating sum.
%Y A345958 Cf. A000097, A027193, A034871, A239830, A341446, A344650, A344651, A344743, A345917, A345918, A345920.
%K A345958 nonn
%O A345958 1,1
%A A345958 _Gus Wiseman_, Jul 11 2021