A359757 Greatest positive integer whose weakly increasing prime indices have zero-based weighted sum (A359674) equal to n.

4, 9, 25, 49, 121, 169, 289, 361, 529, 841, 961, 1369, 1681, 1849, 2209, 2809, 3481, 3721, 4489, 5041, 5329, 6241, 6889, 7921, 9409, 10201, 12167, 11449, 15341, 24389, 16399, 26071, 29791, 31117, 35557, 50653, 39401, 56129, 68921, 58867, 72283, 83521, 79007, 86903, 103823

Offset: 1

Author

Gus Wiseman, Jan 16 2023

Keywords

• nonn

Comments

Appears to first differ from A001248 at a(27) = 12167, A001248(27) = 10609.

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:
    4: {1,1}
    9: {2,2}
   25: {3,3}
   49: {4,4}
  121: {5,5}
  169: {6,6}
  289: {7,7}
  361: {8,8}
  529: {9,9}
  841: {10,10}
		

Links

• Andrew Howroyd, Table of n, a(n) for n = 1..500

Crossrefs

The one-based version is A359497, minimum A359682 (sorted A359755).

Last position of n in A359674, reverse A359677.

The minimum instead of maximum is A359676, sorted A359675, reverse A359681.

A053632 counts compositions by zero-based weighted sum.

A112798 lists prime indices, length A001222, sum A056239, reverse A296150.

A124757 = zero-based weighted sum of standard compositions, reverse A231204.

A304818 gives weighted sums of prime indices, reverse A318283.

A320387 counts multisets by weighted sum, zero-based A359678.

A358136 = partial sums of prime indices, ranked by A358137, reverse A359361.

Cf. A001248, A029931, A055932, A089633, A243055, A358194, A359679, A359680, A359683, A359754.

Programs

Mathematica

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

PARI

a(n)={ my(recurse(r, k, m) = if(k==1, if(m>=r, prime(r)^2),
    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));
  vecmax(vector((sqrtint(8*n+1)-1)\2, k, recurse(n,k,n)));
} \\ Andrew Howroyd, Jan 21 2023

Extensions

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