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

0,2

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.

Links

Table of n, a(n) for n=0..50.

Example

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}

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}]

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.

Sequence in context: A236630 A351381 A342551 * A257851 A133790 A316346

Adjacent sequences: A359678 A359679 A359680 * A359682 A359683 A359684

Keyword

nonn

Author

Gus Wiseman, Jan 15 2023

Status

approved