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

2, 6, 8, 15, 18, 24, 32, 35, 50, 54, 60, 72, 77, 96, 98, 128, 135, 140, 143, 150, 162, 200, 216, 221, 240, 242, 288, 294, 308, 315, 323, 338, 375, 384, 392, 437, 450, 486, 512, 540, 560, 572, 578, 600, 648, 667, 693, 722, 726, 735, 800, 864, 875, 882, 884, 899

Offset: 1

Author

Gus Wiseman, Jul 11 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 one odd conjugate prime index. Conjugate prime indices are listed by A321650, ranked by A122111.

Examples

			The initial terms and their prime indices:
   2: {1}
   6: {1,2}
   8: {1,1,1}
  15: {2,3}
  18: {1,2,2}
  24: {1,1,1,2}
  32: {1,1,1,1,1}
  35: {3,4}
  50: {1,3,3}
  54: {1,2,2,2}
  60: {1,1,2,3}
  72: {1,1,1,2,2}
  77: {4,5}
  96: {1,1,1,1,1,2}
  98: {1,4,4}
		

Crossrefs

The k > 0 version is A000037.

These multisets are counted by A000070.

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

The version for unreversed-alternating sum is A001105.

These partitions are counted by A035363.

These are the positions of 1's in A344616.

The k = 2 version is A345961, counted by A120452.

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

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

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

A025047 counts wiggly compositions.

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).

A316524 gives the alternating sum of prime indices.

A325534 and A325535 count separable and inseparable partitions.

A344606 counts alternating permutations of prime indices.

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

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

Cf. A000097, A027193, A034871, A239830, A341446, A344650, A344651, A344743, A345917, A345918, A345920.

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[#]]==1&]