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A382264 Semiprimes that are the sum of the m-th prime and the m-th semiprime for some m.

%I A382264 #7 Mar 21 2025 11:23:54
%S A382264 6,9,14,25,38,55,86,122,141,158,178,185,218,262,301,326,446,466,537,
%T A382264 634,695,723,758,785,866,878,886,895,898,921,993,1006,1041,1047,1077,
%U A382264 1099,1126,1138,1154,1198,1214,1219,1234,1262,1366,1466,1535,1679,1706,1751,1774,1779,1822,1977,2026,2173
%N A382264 Semiprimes that are the sum of the m-th prime and the m-th semiprime for some m.
%C A382264 Semiprimes in A133796.
%C A382264 Corresponding m's are A092021.
%H A382264 Robert Israel, <a href="/A382264/b382264.txt">Table of n, a(n) for n = 1..10000</a>
%F A382264 a(n) = A133796(A092021(n)).
%e A382264 a(4) = 25 is a term, because 25 = 5^2 is a semiprime and 25 = 11 + 14 where 11 is the 5th prime and 14 is the 5th semiprime.
%p A382264 N:= 10000: # for terms where the m-th prime and m-th semiprime are <= N
%p A382264 P:= select(isprime, [2, seq(i, i=3..N, 2)]): nP:= nops(P):
%p A382264 S:= NULL:
%p A382264 for i from 1 to nP while P[i]^2 <= N do
%p A382264   jmax:= ListTools:-BinaryPlace(P, N/P[i]);
%p A382264   S:= S, op(P[i..jmax] *~ P[i]);
%p A382264 od:
%p A382264 S:= sort([S]):
%p A382264 m:= min(nP, nops(S)):
%p A382264 select(t -> numtheory:-bigomega(t)=2, P[1..m] + S[1..m]);
%Y A382264 Cf. A092021, A133796, A382186.
%K A382264 nonn
%O A382264 1,1
%A A382264 _Zak Seidov_ and _Robert Israel_, Mar 19 2025