cp's OEIS Frontend

A345961 Numbers whose prime indices have reverse-alternating sum 2.

%I A345961 #7 Jul 15 2021 15:08:27
%S A345961 3,10,12,21,27,30,40,48,55,70,75,84,90,91,108,120,147,154,160,187,189,
%T A345961 192,210,220,243,247,250,270,280,286,300,336,360,363,364,391,432,442,
%U A345961 462,480,490,495,507,525,551,588,616,630,640,646,675,713,748,750,756
%N A345961 Numbers whose prime indices have reverse-alternating sum 2.
%C A345961 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 A345961 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 A345961 Also numbers with exactly two odd conjugate prime indices. The restriction to odd omega is A345960, and the restriction to even omega is A345962.
%e A345961 The initial terms and their prime indices:
%e A345961     3: {2}
%e A345961    10: {1,3}
%e A345961    12: {1,1,2}
%e A345961    21: {2,4}
%e A345961    27: {2,2,2}
%e A345961    30: {1,2,3}
%e A345961    40: {1,1,1,3}
%e A345961    48: {1,1,1,1,2}
%e A345961    55: {3,5}
%e A345961    70: {1,3,4}
%e A345961    75: {2,3,3}
%e A345961    84: {1,1,2,4}
%e A345961    90: {1,2,2,3}
%e A345961    91: {4,6}
%e A345961   108: {1,1,2,2,2}
%e A345961   120: {1,1,1,2,3}
%t A345961 primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A345961 sats[y_]:=Sum[(-1)^(i-Length[y])*y[[i]],{i,Length[y]}];
%t A345961 Select[Range[100],sats[primeMS[#]]==2&]
%Y A345961 Below we use k to indicate reverse-alternating sum.
%Y A345961 The k > 0 version is A000037.
%Y A345961 These multisets are counted by A000097.
%Y A345961 The k = 0 version is A000290, counted by A000041.
%Y A345961 These partitions are counted by A120452 (negative: A344741).
%Y A345961 These are the positions of 2's in A344616.
%Y A345961 The k = -1 version is A345912.
%Y A345961 The k = 1 version is A345958.
%Y A345961 The unreversed version is A345960 (negative: A345962).
%Y A345961 A000070 counts partitions with alternating sum 1.
%Y A345961 A002054/A345924/A345923 count/rank compositions with alternating sum -2.
%Y A345961 A027187 counts partitions with reverse-alternating sum <= 0.
%Y A345961 A056239 adds up prime indices,  row sums of A112798.
%Y A345961 A088218/A345925/A345922 count/rank compositions with alternating sum 2.
%Y A345961 A088218 also counts compositions with alternating sum 0, ranked by A344619.
%Y A345961 A097805 counts compositions by alternating (or reverse-alternating) sum.
%Y A345961 A103919 counts partitions by sum and alternating sum (reverse: A344612).
%Y A345961 A316524 gives the alternating sum of prime indices.
%Y A345961 A325534 and A325535 count separable and inseparable partitions.
%Y A345961 A344606 counts alternating permutations of prime indices.
%Y A345961 A344610 counts partitions by sum and positive reverse-alternating sum.
%Y A345961 Cf. A000984, A001791, A025047, A027193, A239830, A341446, A344611, A344650, A344651, A344743, A345910, A345911, A345918.
%K A345961 nonn
%O A345961 1,1
%A A345961 _Gus Wiseman_, Jul 12 2021