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

1,1

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.

Links

Table of n, a(n) for n=1..50.

Example

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}

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&]

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.

Sequence in context: A294366 A110859 A190904 * A271183 A125614 A061936

Adjacent sequences: A345957 A345958 A345959 * A345961 A345962 A345963

Keyword

nonn

Author

Gus Wiseman, Jul 12 2021

Status

approved