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A340444 a(n) is the least prime of the form p*q + p*r + q*r where p is the n-th prime and q and r are primes < p, or 0 if there are none.

%I A340444 #13 Jan 08 2021 04:31:29
%S A340444 0,0,31,41,61,71,151,101,199,151,227,191,211,311,241,271,487,311,479,
%T A340444 653,521,401,421,727,491,823,521,541,773,571,641,661,691,701,751,761,
%U A340444 1109,821,2039,1399,1447,911,1543,971,991,1607,1061,1571,1831,1151,1171,1201,1697,2273,1291,1321,2711
%N A340444 a(n) is the least prime of the form p*q + p*r + q*r where p is the n-th prime and q and r are primes < p, or 0 if there are none.
%C A340444 If prime(k) is in A023219, a(k) = 5*prime(k)+6.
%H A340444 Robert Israel, <a href="/A340444/b340444.txt">Table of n, a(n) for n = 1..2000</a>
%e A340444 a(7) = 151 because prime(7) = 17, and 151 = 17*3+17*5+3*5 is the least prime of the form 17*p + 17*q + p*q.
%p A340444 f:= proc(n) local p,L,i,j,t;
%p A340444   p:= ithprime(n);
%p A340444   L:= sort([seq(seq((ithprime(i)+p)*(ithprime(j)+p)-p^2, i=1..j-1),j=2..n-1)]);
%p A340444   for t in L do if isprime(t) then return t fi od:
%p A340444   0
%p A340444 end proc:
%p A340444 A:= map(f, [$1..100]);
%o A340444 (Python)
%o A340444 from sympy import isprime, prime
%o A340444 def aupto(nn):
%o A340444   alst, plst = [0 for i in range(nn)], [prime(i+1) for i in range(nn)]
%o A340444   for n in range(1, nn+1):
%o A340444     p = plst[n-1]
%o A340444     t = ((p, plst[i], plst[j]) for i in range(n-2) for j in range(i+1, n-1))
%o A340444     for s in sorted(p*q + p*r + q*r for p, q, r in t):
%o A340444       if isprime(s): alst[n-1]=s; break
%o A340444   return alst
%o A340444 print(aupto(57)) # _Michael S. Branicky_, Jan 07 2021
%Y A340444 Cf. A023219, A340439.
%K A340444 nonn,look
%O A340444 1,3
%A A340444 _Robert Israel_, Jan 07 2021