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

3, 10, 12, 21, 27, 30, 40, 48, 55, 70, 75, 84, 90, 91, 108, 120, 147, 154, 160, 187, 189, 192, 210, 220, 243, 247, 250, 270, 280, 286, 300, 336, 360, 363, 364, 391, 432, 442, 462, 480, 490, 495, 507, 525, 551, 588, 616, 630, 640, 646, 675, 713, 748, 750, 756

Offset: 1

Author

Gus Wiseman, Jul 12 2021

Keywords

• nonn

Comments

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.

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.

Also numbers with exactly two odd conjugate prime indices. The restriction to odd omega is A345960, and the restriction to even omega is A345962.

Examples

			The initial terms and their prime indices:
    3: {2}
   10: {1,3}
   12: {1,1,2}
   21: {2,4}
   27: {2,2,2}
   30: {1,2,3}
   40: {1,1,1,3}
   48: {1,1,1,1,2}
   55: {3,5}
   70: {1,3,4}
   75: {2,3,3}
   84: {1,1,2,4}
   90: {1,2,2,3}
   91: {4,6}
  108: {1,1,2,2,2}
  120: {1,1,1,2,3}
		

Crossrefs

Below we use k to indicate reverse-alternating sum.

The k > 0 version is A000037.

These multisets are counted by A000097.

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

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

These are the positions of 2's in A344616.

The k = -1 version is A345912.

The k = 1 version is A345958.

The unreversed version is A345960 (negative: A345962).

A000070 counts partitions with alternating sum 1.

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

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

A056239 adds up prime indices, row sums of A112798.

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

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

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

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

A316524 gives the alternating sum of prime indices.

A325534 and A325535 count separable and inseparable partitions.

A344606 counts alternating permutations of prime indices.

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

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

Programs

Mathematica

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
sats[y_]:=Sum[(-1)^(i-Length[y])*y[[i]],{i,Length[y]}];
Select[Range[100],sats[primeMS[#]]==2&]