A359676 Least positive integer whose weakly increasing prime indices have zero-based weighted sum n (A359674).

1, 4, 6, 8, 14, 12, 16, 20, 30, 24, 32, 36, 40, 52, 48, 56, 100, 72, 80, 92, 96, 104, 112, 124, 136, 148, 176, 152, 214, 172, 184, 188, 262, 212, 272, 236, 248, 244, 304, 268, 346, 284, 328, 292, 386, 316, 398, 332, 376, 356, 458, 388, 478, 404, 472, 412, 526

Offset: 1

Author

Gus Wiseman, Jan 14 2023

Keywords

• nonn
• look

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}
    6: {1,2}
    8: {1,1,1}
   14: {1,4}
   12: {1,1,2}
   16: {1,1,1,1}
   20: {1,1,3}
   30: {1,2,3}
   24: {1,1,1,2}
   32: {1,1,1,1,1}
   36: {1,1,2,2}
   40: {1,1,1,3}
   52: {1,1,6}
   48: {1,1,1,1,2}
		

Crossrefs

First position of n in A359674, reverse A359677.

The sorted version is A359675, reverse A359680.

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

The reverse version is A359681.

The one-based version is A359682, sorted A359755.

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 sums 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, 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[primeMS[n]],{n,1,Prime[nn]^2}];
Table[Position[seq,k][[1,1]],{k,0,nn}]