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A304818 If n = Product_i p(y_i) where p(i) is the i-th prime number and y_i <= y_j for i < j, then a(n) = Sum_i y_i*i. 53

%I #18 May 29 2018 19:17:15

%S 0,1,2,3,3,5,4,6,6,7,5,9,6,9,8,10,7,11,8,12,10,11,9,14,9,13,12,15,10,

%T 14,11,15,12,15,11,17,12,17,14,18,13,17,14,18,15,19,15,20,12,16,16,21,

%U 16,19,13,22,18,21,17,21,18,23,18,21,15,20,19,24,20,19

%N If n = Product_i p(y_i) where p(i) is the i-th prime number and y_i <= y_j for i < j, then a(n) = Sum_i y_i*i.

%C If n > 1 is not a prime number, we have a(n) >= A056239(n) >= Omega(n) >= omega(n) >= A071625(n) >= ... >= Omicron(n) >= omicron(n) > 1, where Omega = A001222, omega = A001221, Omicron = A304687 and omicron = A304465.

%H Alois P. Heinz, <a href="/A304818/b304818.txt">Table of n, a(n) for n = 1..20000</a>

%F a(n) = A056239(A304660(n)).

%e The multiset of prime indices (see A112798) of 216 is {1,1,1,2,2,2}, which becomes {1,2,3,4,4,5,5,6,6} under A304660, so a(216) = 1+2+3+4+4+5+5+6+6 = 36.

%p a:= n-> (l-> add(i*numtheory[pi](l[i]), i=1..nops(l)))(

%p sort(map(i-> i[1]$i[2], ifactors(n)[2]))):

%p seq(a(n), n=1..100); # _Alois P. Heinz_, May 20 2018

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

%t Table[With[{y=primeMS[n]},Sum[y[[i]]*i,{i,Length[y]}]],{n,20}]

%o (PARI) a(n) = {my(f = factor(n), s = 0, i = 0); for (k=1, #f~, for (kk = 1, f[k, 2], i++; s += i*primepi(f[k,1]););); s;} \\ _Michel Marcus_, May 19 2018

%o (PARI) vf(n) = {my(f=factor(n), nb = bigomega(n), g = vector(nb), i = 0); for (k=1, #f~, for (kk = 1, f[k, 2], i++; g[i] = primepi(f[k,1]););); return(g);} \\ A112798

%o a(n) = {my(g = vf(n)); sum(k=1, #g, k*g[k]);} \\ _Michel Marcus_, May 19 2018

%Y Cf. A000720, A001221, A001222, A001358, A055932, A056239, A071625, A112798, A181819, A182850, A182857, A275870, A304465, A304660.

%K nonn,look

%O 1,3

%A _Gus Wiseman_, May 18 2018

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