A359681 Least positive integer whose reversed (weakly decreasing) prime indices have zero-based weighted sum (A359677) equal to n.

1, 4, 9, 8, 18, 50, 16, 36, 100, 54, 32, 72, 81, 108, 300, 64, 144, 400, 216, 600, 243, 128, 288, 800, 432, 486, 1350, 648, 256, 576, 729, 864, 2400, 3375, 1296, 3600, 512, 1152, 1944, 1728, 4800, 9000, 2187, 2916, 8100, 1024, 2304, 6400, 3456, 4374, 12150

Offset: 0

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}
    9: {2,2}
    8: {1,1,1}
   18: {1,2,2}
   50: {1,3,3}
   16: {1,1,1,1}
   36: {1,1,2,2}
  100: {1,1,3,3}
   54: {1,2,2,2}
   32: {1,1,1,1,1}
   72: {1,1,1,2,2}
   81: {2,2,2,2}
  108: {1,1,2,2,2}
  300: {1,1,2,3,3}
		

Crossrefs

The unreversed version is A359676.

First position of n in A359677, reverse A359674.

The one-based version is A359679, sorted A359754.

The sorted version is A359680, reverse A359675.

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

A053632 counts compositions by zero-based weighted sum.

A112798 lists prime indices, length A001222, sum A056239.

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

A304818 gives weighted sum of prime indices, reverse A318283.

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

Cf. A001248, A029931, A055932, A089633, A243055, A359043, A358194, A359360, A359361, A359497, A359683.

Programs

Mathematica

nn=20;
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,Prime[nn]^2}];
Table[Position[seq,k][[1,1]],{k,0,nn}]