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Least positive integer whose weakly increasing prime indices have zero-based weighted sum n (A359674).
14

%I #6 Jan 15 2023 09:51:05

%S 1,4,6,8,14,12,16,20,30,24,32,36,40,52,48,56,100,72,80,92,96,104,112,

%T 124,136,148,176,152,214,172,184,188,262,212,272,236,248,244,304,268,

%U 346,284,328,292,386,316,398,332,376,356,458,388,478,404,472,412,526

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

%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.

%e The terms together with their prime indices begin:

%e 1: {}

%e 4: {1,1}

%e 6: {1,2}

%e 8: {1,1,1}

%e 14: {1,4}

%e 12: {1,1,2}

%e 16: {1,1,1,1}

%e 20: {1,1,3}

%e 30: {1,2,3}

%e 24: {1,1,1,2}

%e 32: {1,1,1,1,1}

%e 36: {1,1,2,2}

%e 40: {1,1,1,3}

%e 52: {1,1,6}

%e 48: {1,1,1,1,2}

%t nn=20;

%t primeMS[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[primeMS[n]],{n,1,Prime[nn]^2}];

%t Table[Position[seq,k][[1,1]],{k,0,nn}]

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

%Y The sorted version is A359675, reverse A359680.

%Y The reverse one-based version is A359679, sorted A359754.

%Y The reverse version is A359681.

%Y The one-based version is A359682, sorted A359755.

%Y The version for standard compositions is A359756, one-based A089633.

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

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

%Y A124757 gives zero-based weighted sum of standard compositions, rev A231204.

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

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

%Y A358136 lists partial sums of prime indices, ranked by A358137, rev A359361.

%Y Cf. A001248, A029931, A055932, A243055, A359043, A358194, A359497, A359683.

%K nonn,look

%O 1,2

%A _Gus Wiseman_, Jan 14 2023