A359675 Positions of first appearances in the sequence of zero-based weighted sums of prime indices (A359674).

1, 4, 6, 8, 12, 14, 16, 20, 24, 30, 32, 36, 40, 48, 52, 56, 72, 80, 92, 96, 100, 104, 112, 124, 136, 148, 152, 172, 176, 184, 188, 212, 214, 236, 244, 248, 262, 268, 272, 284, 292, 304, 316, 328, 332, 346, 356, 376, 386, 388, 398, 404, 412, 428, 436, 452, 458

Offset

1,2

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=1..57.

Example

The terms together with their prime indices begin:
   1: {}
   4: {1,1}
   6: {1,2}
   8: {1,1,1}
  12: {1,1,2}
  14: {1,4}
  16: {1,1,1,1}
  20: {1,1,3}
  24: {1,1,1,2}
  30: {1,2,3}
  32: {1,1,1,1,1}
  36: {1,1,2,2}
  40: {1,1,1,3}
  48: {1,1,1,1,2}

Mathematica

nn=100;
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[primeMS[n]], {n, 1, nn}];
Select[Range[nn], FreeQ[seq[[Range[#-1]]], seq[[#]]]&]

Crossrefs

Positions of first appearances in A359674.

The unsorted version A359676.

The reverse version is A359680, unsorted A359681.

The reverse one-based version is A359754, unsorted A359679.

The 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.

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.

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

Cf. A001248, A029931, A055932, A243055, A359043, A358194, A359360, A359497, A359677, A359683.

Sequence in context: A249722 A047407 A090697 * A225508 A235036 A107303

Adjacent sequences: A359672 A359673 A359674 * A359676 A359677 A359678

Keyword

nonn

Author

Gus Wiseman, Jan 13 2023

Status

approved