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Greatest positive integer whose weakly increasing prime indices have zero-based weighted sum (A359674) equal to n.
2

%I #12 Jan 21 2023 22:27:00

%S 4,9,25,49,121,169,289,361,529,841,961,1369,1681,1849,2209,2809,3481,

%T 3721,4489,5041,5329,6241,6889,7921,9409,10201,12167,11449,15341,

%U 24389,16399,26071,29791,31117,35557,50653,39401,56129,68921,58867,72283,83521,79007,86903,103823

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

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

%C 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 The zero-based weighted sum of a sequence (y_1,...,y_k) is Sum_{i=1..k} (i-1)*y_i.

%H Andrew Howroyd, <a href="/A359757/b359757.txt">Table of n, a(n) for n = 1..500</a>

%e The terms together with their prime indices begin:

%e 4: {1,1}

%e 9: {2,2}

%e 25: {3,3}

%e 49: {4,4}

%e 121: {5,5}

%e 169: {6,6}

%e 289: {7,7}

%e 361: {8,8}

%e 529: {9,9}

%e 841: {10,10}

%t nn=10;

%t prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];

%t wts[y_]:=Sum[(i-1)*y[[i]],{i,Length[y]}];

%t seq=Table[wts[prix[n]],{n,2^nn}];

%t Table[Position[seq,k][[-1,1]],{k,nn}]

%o (PARI) a(n)={ my(recurse(r, k, m) = if(k==1, if(m>=r, prime(r)^2),

%o 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 vecmax(vector((sqrtint(8*n+1)-1)\2, k, recurse(n,k,n)));

%o } \\ _Andrew Howroyd_, Jan 21 2023

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

%Y Last position of n in A359674, reverse A359677.

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

%Y A053632 counts compositions by zero-based weighted sum.

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

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

%Y A304818 gives weighted sums of prime indices, reverse A318283.

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

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

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

%K nonn

%O 1,1

%A _Gus Wiseman_, Jan 16 2023

%E Terms a(21) and beyond from _Andrew Howroyd_, Jan 21 2023