cp's OEIS Frontend

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

%I A359757 #12 Jan 21 2023 22:27:00
%S A359757 4,9,25,49,121,169,289,361,529,841,961,1369,1681,1849,2209,2809,3481,
%T A359757 3721,4489,5041,5329,6241,6889,7921,9409,10201,12167,11449,15341,
%U A359757 24389,16399,26071,29791,31117,35557,50653,39401,56129,68921,58867,72283,83521,79007,86903,103823
%N A359757 Greatest positive integer whose weakly increasing prime indices have zero-based weighted sum (A359674) equal to n.
%C A359757 Appears to first differ from A001248 at a(27) = 12167, A001248(27) = 10609.
%C A359757 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 A359757 The zero-based weighted sum of a sequence (y_1,...,y_k) is Sum_{i=1..k} (i-1)*y_i.
%H A359757 Andrew Howroyd, <a href="/A359757/b359757.txt">Table of n, a(n) for n = 1..500</a>
%e A359757 The terms together with their prime indices begin:
%e A359757     4: {1,1}
%e A359757     9: {2,2}
%e A359757    25: {3,3}
%e A359757    49: {4,4}
%e A359757   121: {5,5}
%e A359757   169: {6,6}
%e A359757   289: {7,7}
%e A359757   361: {8,8}
%e A359757   529: {9,9}
%e A359757   841: {10,10}
%t A359757 nn=10;
%t A359757 prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A359757 wts[y_]:=Sum[(i-1)*y[[i]],{i,Length[y]}];
%t A359757 seq=Table[wts[prix[n]],{n,2^nn}];
%t A359757 Table[Position[seq,k][[-1,1]],{k,nn}]
%o A359757 (PARI) a(n)={ my(recurse(r, k, m) = if(k==1, if(m>=r, prime(r)^2),
%o A359757     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 A359757   vecmax(vector((sqrtint(8*n+1)-1)\2, k, recurse(n,k,n)));
%o A359757 } \\ _Andrew Howroyd_, Jan 21 2023
%Y A359757 The one-based version is A359497, minimum A359682 (sorted A359755).
%Y A359757 Last position of n in A359674, reverse A359677.
%Y A359757 The minimum instead of maximum is A359676, sorted A359675, reverse A359681.
%Y A359757 A053632 counts compositions by zero-based weighted sum.
%Y A359757 A112798 lists prime indices, length A001222, sum A056239, reverse A296150.
%Y A359757 A124757 = zero-based weighted sum of standard compositions, reverse A231204.
%Y A359757 A304818 gives weighted sums of prime indices, reverse A318283.
%Y A359757 A320387 counts multisets by weighted sum, zero-based A359678.
%Y A359757 A358136 = partial sums of prime indices, ranked by A358137, reverse A359361.
%Y A359757 Cf. A001248, A029931, A055932, A089633, A243055, A358194, A359679, A359680, A359683, A359754.
%K A359757 nonn
%O A359757 1,1
%A A359757 _Gus Wiseman_, Jan 16 2023
%E A359757 Terms a(21) and beyond from _Andrew Howroyd_, Jan 21 2023