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A359683 Greatest positive integer whose reversed (weakly decreasing) prime indices have weighted sum (A318283) equal to n.

%I A359683 #12 Jan 28 2023 12:15:05
%S A359683 1,2,3,5,7,11,14,22,26,34,44,55,68,85,110,130,170,190,242,290,374,418,
%T A359683 506,638,748,836,1012,1276,1364,1628,1914,2090,2552,3190,3410,4070,
%U A359683 4510,5060,6380,7018,8140,9020,9922,11396,14036,15004,17908,19844,21692,23452
%N A359683 Greatest positive integer whose reversed (weakly decreasing) prime indices have weighted sum (A318283) equal to n.
%C A359683 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 A359683 The weighted sum of a sequence (y_1,...,y_k) is Sum_{i=1..k} i*y_i.
%e A359683 The terms together with their prime indices begin:
%e A359683       1: {}
%e A359683       2: {1}
%e A359683       3: {2}
%e A359683       5: {3}
%e A359683       7: {4}
%e A359683      11: {5}
%e A359683      14: {1,4}
%e A359683      22: {1,5}
%e A359683      26: {1,6}
%e A359683      34: {1,7}
%e A359683      44: {1,1,5}
%e A359683      55: {3,5}
%e A359683      68: {1,1,7}
%e A359683      85: {3,7}
%e A359683     110: {1,3,5}
%e A359683     130: {1,3,6}
%e A359683     170: {1,3,7}
%e A359683     190: {1,3,8}
%e A359683     242: {1,5,5}
%e A359683     290: {1,3,10}
%e A359683 The 6 numbers with weighted sum of reversed prime indices 9, together with their prime indices:
%e A359683   18: {1,2,2}
%e A359683   23: {9}
%e A359683   25: {3,3}
%e A359683   28: {1,1,4}
%e A359683   33: {2,5}
%e A359683   34: {1,7}
%e A359683 Hence a(9) = 34.
%t A359683 nn=10;
%t A359683 primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A359683 ots[y_]:=Sum[i*y[[i]],{i,Length[y]}];
%t A359683 seq=Table[ots[Reverse[primeMS[n]]],{n,1,2^nn}];
%t A359683 Table[Position[seq,k][[-1,1]],{k,0,nn}]
%Y A359683 First position of n in A318283, unreversed A304818.
%Y A359683 The unreversed version is A359497.
%Y A359683 The least instead of greatest is A359679, unreversed A359682.
%Y A359683 A112798 lists prime indices, length A001222, sum A056239.
%Y A359683 A320387 counts multisets by weighted sum, zero-based A359678.
%Y A359683 A358136 lists partial sums of prime indices, ranked by A358137, rev A359361.
%Y A359683 Cf. A001248, A029931, A055932, A089633, A243055, A358194, A359043, A359676, A359681, A359755.
%K A359683 nonn
%O A359683 0,2
%A A359683 _Gus Wiseman_, Jan 15 2023
%E A359683 More terms from _Jinyuan Wang_, Jan 26 2023