A345961 - OEIS

%I #7 Jul 15 2021 15:08:27

%S 3,10,12,21,27,30,40,48,55,70,75,84,90,91,108,120,147,154,160,187,189,

%T 192,210,220,243,247,250,270,280,286,300,336,360,363,364,391,432,442,

%U 462,480,490,495,507,525,551,588,616,630,640,646,675,713,748,750,756

%N Numbers whose prime indices have reverse-alternating sum 2.

%C 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 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 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 The initial terms and their prime indices:

%e     3: {2}

%e    10: {1,3}

%e    12: {1,1,2}

%e    21: {2,4}

%e    27: {2,2,2}

%e    30: {1,2,3}

%e    40: {1,1,1,3}

%e    48: {1,1,1,1,2}

%e    55: {3,5}

%e    70: {1,3,4}

%e    75: {2,3,3}

%e    84: {1,1,2,4}

%e    90: {1,2,2,3}

%e    91: {4,6}

%e   108: {1,1,2,2,2}

%e   120: {1,1,1,2,3}

%t primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];

%t sats[y_]:=Sum[(-1)^(i-Length[y])*y[[i]],{i,Length[y]}];

%t Select[Range[100],sats[primeMS[#]]==2&]

%Y Below we use k to indicate reverse-alternating sum.

%Y The k > 0 version is A000037.

%Y These multisets are counted by A000097.

%Y The k = 0 version is A000290, counted by A000041.

%Y These partitions are counted by A120452 (negative: A344741).

%Y These are the positions of 2's in A344616.

%Y The k = -1 version is A345912.

%Y The k = 1 version is A345958.

%Y The unreversed version is A345960 (negative: A345962).

%Y A000070 counts partitions with alternating sum 1.

%Y A002054/A345924/A345923 count/rank compositions with alternating sum -2.

%Y A027187 counts partitions with reverse-alternating sum <= 0.

%Y A056239 adds up prime indices,  row sums of A112798.

%Y A088218/A345925/A345922 count/rank compositions with alternating sum 2.

%Y A088218 also counts compositions with alternating sum 0, ranked by A344619.

%Y A097805 counts compositions by alternating (or reverse-alternating) sum.

%Y A103919 counts partitions by sum and alternating sum (reverse: A344612).

%Y A316524 gives the alternating sum of prime indices.

%Y A325534 and A325535 count separable and inseparable partitions.

%Y A344606 counts alternating permutations of prime indices.

%Y A344610 counts partitions by sum and positive reverse-alternating sum.

%Y Cf. A000984, A001791, A025047, A027193, A239830, A341446, A344611, A344650, A344651, A344743, A345910, A345911, A345918.

%K nonn

%O 1,1

%A _Gus Wiseman_, Jul 12 2021