A359682 Least positive integer whose weakly increasing prime indices have weighted sum (A304818) equal to n.

1, 2, 3, 4, 7, 6, 8, 10, 15, 12, 16, 18, 20, 26, 24, 28, 50, 36, 40, 46, 48, 52, 56, 62, 68, 74, 88, 76, 107, 86, 92, 94, 131, 106, 136, 118, 124, 122, 152, 134, 173, 142, 164, 146, 193, 158, 199, 166, 188, 178, 229, 194, 239, 202, 236, 206, 263, 214, 271, 218

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

Links

Table of n, a(n) for n=0..59.

Example

The 5 numbers with weighted sum of prime indices 12, together with their prime indices:
  20: {1,1,3}
  27: {2,2,2}
  33: {2,5}
  37: {12}
  49: {4,4}
Hence a(12) = 20.

Mathematica

nn=20;
primeMS[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];
ots[y_]:=Sum[i*y[[i]], {i, Length[y]}];
seq=Table[ots[primeMS[n]], {n, 1, Prime[nn]^2}];
Table[Position[seq, k][[1, 1]], {k, 0, nn}]

Crossrefs

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

First position of n in A304818, reverse A318283.

The greatest instead of least is A359497, reverse A359683.

The sorted zero-based version is A359675, reverse A359680.

The zero-based version is A359676, reverse A359681.

The reverse version is A359679.

The sorted version is A359755, reverse A359754.

A112798 lists prime indices, length A001222, sum A056239.

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.

Sequence in context: A210202 A358201 A358176 * A260436 A255553 A260742

Adjacent sequences: A359679 A359680 A359681 * A359683 A359684 A359685

Keyword

nonn

Author

Gus Wiseman, Jan 15 2023

Status

approved