A359680 Positions of first appearances in the sequence of zero-based weighted sums of reversed prime indices (A359677).

1, 4, 8, 9, 16, 18, 32, 36, 50, 54, 64, 72, 81, 100, 108, 128, 144, 216, 243, 256, 288, 300, 400, 432, 486, 512, 576, 600, 648, 729, 800, 864, 1024, 1152, 1296, 1350, 1728, 1944, 2048, 2187, 2304, 2400, 2916, 3375, 3456, 3600, 4096, 4374, 4608, 4800, 5184

Offset: 1

Author

Gus Wiseman, Jan 15 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.

The zero-based weighted sum of a sequence (y_1,...,y_k) is Sum_{i=1..k} (i-1)*y_i.

Examples

			The terms together with their prime indices begin:
     1: {}
     4: {1,1}
     8: {1,1,1}
     9: {2,2}
    16: {1,1,1,1}
    18: {1,2,2}
    32: {1,1,1,1,1}
    36: {1,1,2,2}
    50: {1,3,3}
    54: {1,2,2,2}
    64: {1,1,1,1,1,1}
    72: {1,1,1,2,2}
    81: {2,2,2,2}
   100: {1,1,3,3}
   108: {1,1,2,2,2}
   128: {1,1,1,1,1,1,1}
		

Crossrefs

The unreversed version is A359675, unsorted A359676.

Positions of first appearances in A359677, unreversed A359674.

This is the sorted version of A359681.

The one-based version is A359754, unsorted A359679.

The unreversed one-based version is A359755, unsorted A359682.

The version for standard compositions is A359756, one-based A089633.

A053632 counts compositions by zero-based weighted sum.

A112798 lists prime indices, length A001222, sum A056239, reverse A296150.

A124757 gives zero-based weighted sums of standard compositions, rev A231204.

A304818 gives weighted sum of prime indices, reverse A318283.

A320387 counts multisets by weighted sum, zero-based A359678.

A358136 lists partial sums of prime indices, ranked by A358137, rev A359361.

Cf. A029931, A055932, A243055, A358194, A359043, A359683.

Programs

Mathematica

nn=1000;
primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
wts[y_]:=Sum[(i-1)*y[[i]],{i,Length[y]}];
seq=Table[wts[Reverse[primeMS[n]]],{n,1,nn}];
Select[Range[nn],FreeQ[seq[[Range[#-1]]],seq[[#]]]&]