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A339423 If n = p_1 * ... * p_m with primes p_i <= p_{i+1}, a(n) = Sum_{k<m} Product_{j <= k} p_j. 3

%I

%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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Last modified September 22 19:36 EDT 2021. Contains 347608 sequences. (Running on oeis4.)