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

%I A359497 #13 Jan 21 2023 22:26:51
%S A359497 1,2,3,5,7,11,13,17,19,25,29,35,49,55,77,121,91,143,169,187,221,289,
%T A359497 247,323,361,391,437,539,605,847,1331,715,1001,1573,1183,1859,2197,
%U A359497 1547,2431,2873,3179,3757,4913,3553,4199,5491,4693,6137,6859,9317,14641
%N A359497 Greatest positive integer whose weakly increasing prime indices have weighted sum (A304818) equal to n.
%C A359497 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.
%C A359497 The weighted sum of a sequence (y_1,...,y_k) is Sum_{i=1..k} i*y_i.
%H A359497 Andrew Howroyd, <a href="/A359497/b359497.txt">Table of n, a(n) for n = 0..500</a>
%e A359497 The terms together with their prime indices begin:
%e A359497     1: {}
%e A359497     2: {1}
%e A359497     3: {2}
%e A359497     5: {3}
%e A359497     7: {4}
%e A359497    11: {5}
%e A359497    13: {6}
%e A359497    17: {7}
%e A359497    19: {8}
%e A359497    25: {3,3}
%e A359497    29: {10}
%e A359497    35: {3,4}
%e A359497    49: {4,4}
%e A359497    55: {3,5}
%e A359497    77: {4,5}
%e A359497 The 5 numbers with weighted sum of prime indices 12, together with their prime indices:
%e A359497   20: {1,1,3}
%e A359497   27: {2,2,2}
%e A359497   33: {2,5}
%e A359497   37: {12}
%e A359497   49: {4,4}
%e A359497 Hence a(12) = 49.
%t A359497 nn=10;
%t A359497 primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A359497 ots[y_]:=Sum[i*y[[i]],{i,Length[y]}];
%t A359497 seq=Table[ots[primeMS[n]],{n,1,2^nn}];
%t A359497 Table[Position[seq,k][[-1,1]],{k,0,nn}]
%o A359497 (PARI)
%o A359497 a(n)={ my(recurse(r, k, m) = if(k==1, if(m>=r, prime(r)),
%o A359497   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));
%o A359497   if(n==0, 1, vecmax(vector((sqrtint(8*n+1)-1)\2, k, recurse(n, k, n))));
%o A359497 } \\ _Andrew Howroyd_, Jan 21 2023
%Y A359497 First position of n in A304818, reverse A318283.
%Y A359497 The least instead of greatest is given by A359682, reverse A359679.
%Y A359497 The reverse version is A359683.
%Y A359497 A112798 lists prime indices, length A001222, sum A056239.
%Y A359497 A320387 counts multisets by weighted sum, zero-based A359678.
%Y A359497 A358136 lists partial sums of prime indices, ranked by A358137, rev A359361.
%Y A359497 Cf. A001248, A029931, A055932, A089633, A243055, A358194, A359043, A359676, A359681, A359755.
%K A359497 nonn
%O A359497 0,2
%A A359497 _Gus Wiseman_, Jan 15 2023
%E A359497 Terms a(21) and beyond from _Andrew Howroyd_, Jan 21 2023