A363621 Positive integers whose prime indices have reverse-weighted alternating sum 0.

1, 6, 21, 40, 50, 54, 65, 132, 133, 154, 210, 224, 319, 340, 351, 360, 374, 392, 450, 481, 486, 507, 546, 598, 624, 644, 731, 825, 855, 969, 1007, 1029, 1054, 1144, 1210, 1254, 1320, 1364, 1386, 1403, 1408, 1520, 1558, 1653, 1750, 1785, 1827, 1836, 1890, 1960

Offset: 1

Author

Gus Wiseman, Jun 13 2023

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.

We define the reverse-weighted alternating sum of a sequence (y_1,...,y_k) to be Sum_{i=1..k} (-1)^(k-i) i * y_{k-i+1}.

Examples

			The prime indices of 360 are {1,1,1,2,2,3}, with reverse-weighted alternating sum 1*3 - 2*2 + 3*2 - 4*1 + 5*1 - 6*1 = 0, so 360 is in the sequence.
The terms together with their prime indices begin:
     1: {}
     6: {1,2}
    21: {2,4}
    40: {1,1,1,3}
    50: {1,3,3}
    54: {1,2,2,2}
    65: {3,6}
   132: {1,1,2,5}
   133: {4,8}
   154: {1,4,5}
   210: {1,2,3,4}
   224: {1,1,1,1,1,4}
   319: {5,10}
   340: {1,1,3,7}
   351: {2,2,2,6}
   360: {1,1,1,2,2,3}
		

Crossrefs

The unweighted version is A000290.

Partitions of this type are counted by A363532.

Positions of zeros in A363620 and A363624, reverse A363619 and A363625.

Compositions of this type are counted by A363626.

A053632 counts compositions by weighted sum.

A055396 gives minimum prime index, maximum A061395.

A112798 lists prime indices, length A001222, sum A056239.

A264034 counts partitions by weighted sum, reverse A358194.

A304818 gives weighted sum of prime indices.

A318283 gives weighted sum of reversed prime indices.

A320387 counts multisets by weighted sum.

A344616 gives reverse-alternating sum of prime indices.

A363623 counts partitions by reverse-weighted alternating sum.

Cf. A046660, A106529, A124010, A181819, A261079, A316524, A359674, A363622.

Programs

Mathematica

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
revaltwtsum[y_]:=Sum[(-1)^(Length[y]-k)*k*y[[-k]],{k,1,Length[y]}];
Select[Range[1000],revaltwtsum[prix[#]]==0&]