A345960 Numbers whose prime indices have alternating sum 2.

3, 12, 27, 30, 48, 70, 75, 108, 120, 147, 154, 192, 243, 270, 280, 286, 300, 363, 432, 442, 480, 507, 588, 616, 630, 646, 675, 750, 768, 867, 874, 972, 1080, 1083, 1120, 1144, 1200, 1323, 1334, 1386, 1452, 1470, 1587, 1728, 1750, 1768, 1798, 1875, 1920, 2028

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 odd Omega (A001222) and exactly two odd conjugate prime indices. The version for even Omega is A345962, and the union is A345961. Conjugate prime indices are listed by A321650 and ranked by A122111.

Examples

			The initial terms and their prime indices:
    3: {2}
   12: {1,1,2}
   27: {2,2,2}
   30: {1,2,3}
   48: {1,1,1,1,2}
   70: {1,3,4}
   75: {2,3,3}
  108: {1,1,2,2,2}
  120: {1,1,1,2,3}
  147: {2,4,4}
  154: {1,4,5}
  192: {1,1,1,1,1,1,2}
  243: {2,2,2,2,2}
  270: {1,2,2,2,3}
  280: {1,1,1,3,4}
  286: {1,5,6}
  300: {1,1,2,3,3}
		

Crossrefs

These partitions are counted by A000097.

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

The k = 1 version is A001105 (reverse: A345958).

The k > 0 version is A026424.

These multisets are counted by A120452.

These are the positions of 2's in A316524 (reverse: A344616).

The k = -1 version is A345959.

The version for reversed alternating sum is A345961.

The k = -2 version is A345962.

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

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

A056239 adds up prime indices, row sums of A112798.

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

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

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

A325534 and A325535 count separable and inseparable partitions.

A344606 counts alternating permutations of prime indices.

Cf. A000037, A000070, A028260, A035363, A239830, A341446, A344609, A344610, A344651, A344741, A345197.

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[#]]==2&]