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A382186 Prime numbers that are the sum of the m-th prime and the m-th semiprime for some m.

%I A382186 #12 Mar 20 2025 13:59:08
%S A382186 17,41,71,131,281,331,353,397,449,487,563,953,1279,1289,1409,1627,
%T A382186 2621,2999,3533,3631,3697,3989,4057,4133,4523,4603,4733,4919,5273,
%U A382186 5591,5641,6211,6247,6269,6299,6469,6803,7753,7879,7937,8353,8543,8971,8999,9041,9181,9413,9479,9787,9887,9941,10487
%N A382186 Prime numbers that are the sum of the m-th prime and the m-th semiprime for some m.
%C A382186 Primes in A133796.
%C A382186 Corresponding m's are in A092108.
%H A382186 Robert Israel, <a href="/A382186/b382186.txt">Table of n, a(n) for n = 1..10000</a>
%F A382186 a(n) = A133796(A092108(n)).
%e A382186 a(3) = 71 is a term because 71 = 37 + 34 is prime, where 37 is the 12th prime and 34 is the 12th semiprime.
%p A382186 N:= 10000: # for terms where the m-th prime and m-th semiprime are <= N
%p A382186 P:= select(isprime, [2,seq(i,i=3..N,2)]): nP:= nops(P):
%p A382186 S:= NULL:
%p A382186 for i from 1 to nP while P[i]^2 <= N do
%p A382186   jmax:= ListTools:-BinaryPlace(P, N/P[i]);
%p A382186   S:= S, op(P[i..jmax] *~ P[i]);
%p A382186 od:
%p A382186 S:= sort([S]):
%p A382186 m:= min(nP, nops(S)):
%p A382186 select(isprime,P[1..m] + S[1..m]);
%t A382186 sp=Select[Range[3300],PrimeOmega[#]==2&];p=Prime[Range[Length[sp]]];Select[p+sp,PrimeQ] (* _James C. McMahon_, Mar 20 2025 *)
%Y A382186 Cf. A000040, A001358, A092108, A133796.
%K A382186 nonn
%O A382186 1,1
%A A382186 _Zak Seidov_ and _Robert Israel_, Mar 18 2025