A359497 Greatest positive integer whose weakly increasing prime indices have weighted sum (A304818) equal to n.

1, 2, 3, 5, 7, 11, 13, 17, 19, 25, 29, 35, 49, 55, 77, 121, 91, 143, 169, 187, 221, 289, 247, 323, 361, 391, 437, 539, 605, 847, 1331, 715, 1001, 1573, 1183, 1859, 2197, 1547, 2431, 2873, 3179, 3757, 4913, 3553, 4199, 5491, 4693, 6137, 6859, 9317, 14641

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

Andrew Howroyd, Table of n, a(n) for n = 0..500

Example

The terms together with their prime indices begin:
    1: {}
    2: {1}
    3: {2}
    5: {3}
    7: {4}
   11: {5}
   13: {6}
   17: {7}
   19: {8}
   25: {3,3}
   29: {10}
   35: {3,4}
   49: {4,4}
   55: {3,5}
   77: {4,5}
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) = 49.

Mathematica

nn=10;
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, 2^nn}];
Table[Position[seq, k][[-1, 1]], {k, 0, nn}]

Prog

(PARI)
a(n)={ my(recurse(r, k, m) = if(k==1, if(m>=r, prime(r)),
  my(z=0); for(j=1, min(m, (r-k*(k-1)/2)\k), z=max(z, self()(r-k*j, k-1, j)*prime(j))); z));
  if(n==0, 1, vecmax(vector((sqrtint(8*n+1)-1)\2, k, recurse(n, k, n))));
} \\ Andrew Howroyd, Jan 21 2023

Crossrefs

First position of n in A304818, reverse A318283.

The least instead of greatest is given by A359682, reverse A359679.

The reverse version is A359683.

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, A089633, A243055, A358194, A359043, A359676, A359681, A359755.

Sequence in context: A341667 A330007 A008792 * A094746 A049543 A294200

Adjacent sequences: A359494 A359495 A359496 * A359498 A359499 A359500

Keyword

nonn

Author

Gus Wiseman, Jan 15 2023

Extensions

Terms a(21) and beyond from Andrew Howroyd, Jan 21 2023

Status

approved