A345959 Numbers whose prime indices have alternating sum -1.

6, 15, 24, 35, 54, 60, 77, 96, 135, 140, 143, 150, 216, 221, 240, 294, 308, 315, 323, 375, 384, 437, 486, 540, 560, 572, 600, 667, 693, 726, 735, 864, 875, 884, 899, 960, 1014, 1147, 1176, 1215, 1232, 1260, 1287, 1292, 1350, 1500, 1517, 1536, 1715, 1734, 1748

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 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.

Also numbers with even Omega (A001222) and exactly one odd conjugate prime index. Conjugate prime indices are listed by A321650, ranked by A122111.

Examples

			The initial terms and their prime indices:
    6: {1,2}
   15: {2,3}
   24: {1,1,1,2}
   35: {3,4}
   54: {1,2,2,2}
   60: {1,1,2,3}
   77: {4,5}
   96: {1,1,1,1,1,2}
  135: {2,2,2,3}
  140: {1,1,3,4}
  143: {5,6}
  150: {1,2,3,3}
  216: {1,1,1,2,2,2}
  221: {6,7}
  240: {1,1,1,1,2,3}
		

Crossrefs

These multisets are counted by A000070.

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

The k = 1 version is A001105.

The k > 0 version is A026424.

These are the positions of -1's in A316524.

The k = 2 version is A345960.

The k = -2 version is A345962.

A000984/A345909/A345911 count/rank compositions with alternating sum 1.

A001791/A345910/A345912 count/rank compositions with alternating sum -1.

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

A056239 adds up prime indices, row sums of A112798.

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

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

A325534/A325535 count separable/inseparable partitions.

A344607 counts partitions with reverse-alternating sum >= 0.

A344616 gives the alternating sum of reversed prime indices.

Cf. A000097, A028260, A035363, A236913, A239830, A341446, A344609, A344610, A344651, A345919, A345961.

Programs

Mathematica

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
ats[y_]:=Sum[(-1)^(i-1)*y[[i]],{i,Length[y]}];
Select[Range[0,100],ats[primeMS[#]]==-1&]