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A382831 a(n) is the n-th n-almost-prime that is a partial sum of the sequence of n-almost-primes.

%I A382831 #16 Apr 29 2025 13:27:31
%S A382831 2,10,964,1804,7820,48120,830817,4895208,11308160,162802560,394129476,
%T A382831 3763612800,19823090472,1018716103620,9744542956800,3989325082624,
%U A382831 329306801920000,2978224618328064,11804664377696256,128906665137012736
%N A382831 a(n) is the n-th n-almost-prime that is a partial sum of the sequence of n-almost-primes.
%e A382831 The first three members of A086062 that are 3-almost-primes are 8 = 2^3, 20 = 2^2 * 5 = 8 + 12, and 964 = 2^2 * 241 = 8 + 12 + 18 + ... + 92, so a(3) = 964.
%p A382831 f:= proc(n) uses priqueue;
%p A382831   local pq,t,s,x,p,i,count;
%p A382831   initialize(pq);
%p A382831   insert([-2^n,2$n],pq);
%p A382831   s:= 0; count:= 0:
%p A382831   do
%p A382831     t:= extract(pq);
%p A382831     x:= -t[1];
%p A382831     s:= s + x;
%p A382831     if numtheory:-bigomega(s) = n  then count:= count+1; if count = n then return s fi fi;
%p A382831     p:= nextprime(t[-1]);
%p A382831     for i from n+1 to 2 by -1 while t[i] = t[-1] do
%p A382831           insert([t[1]*(p/t[-1])^(n+2-i), op(t[2..i-1]), p$(n+2-i)], pq)
%p A382831     od;
%p A382831   od
%p A382831 end proc:
%p A382831 map(f, [$1..20]);
%Y A382831 Cf. A007504, A062198, A086062, A086046, A086047,  A086052, A086059, A086061.
%K A382831 nonn
%O A382831 1,1
%A A382831 _Robert Israel_, Apr 28 2025