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%I #12 Dec 06 2020 18:42:37
%S 0,0,0,2,0,2,0,6,3,2,0,6,0,2,3,14,0,8,0,6,3,2,0,14,5,2,12,6,0,8,0,30,
%T 3,2,5,18,0,2,3,14,0,8,0,6,12,2,0,30,7,12,3,6,0,26,5,14,3,2,0,18,0,2,
%U 12,62,5,8,0,6,3,12,0,38,0,2,18,6,7,8,0,30,39,2,0,18,5,2,3,14,0,26,7,6
%N If n = p_1 * ... * p_m with primes p_i <= p_{i+1}, a(n) = Sum_{k<m} Product_{j <= k} p_j.
%C If n is prime, a(n)=0.
%C a(n) is odd if and only if n is odd and A001222(n) is even.
%H Robert Israel, <a href="/A339423/b339423.txt">Table of n, a(n) for n = 1..10000</a>
%e 90 = 2*3*3*5 so a(90) = 2 + 2*3 + 2*3*3 = 26.
%p f:= proc(n)
%p local F,T,P,j;
%p F:= sort(map(t -> t[1]$t[2], ifactors(n)[2]));
%p T:= 0; P:= 1;
%p for j from 1 to nops(F)-1 do
%p P:= P*F[j];
%p T:= T+P;
%p od;
%p T
%p end proc:
%p map(f, [$1..200]);
%o (PARI) conv(n) = {my(f=factor(n), v=vector(bigomega(n)), k=1); for (i=1, #f~, for (j=1, f[i,2], v[k] = f[i,1]; k++;);); v;}
%o a(n) = {my(v=conv(n)); sum(k=1, #v-1, prod(j=1, k, v[j]));} \\ _Michel Marcus_, Dec 04 2020
%Y Cf. A001222.
%K nonn
%O 1,4
%A _J. M. Bergot_ and _Robert Israel_, Dec 03 2020
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