A133796 -id:A133796 - cp's OEIS frontend

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

17, 41, 71, 131, 281, 331, 353, 397, 449, 487, 563, 953, 1279, 1289, 1409, 1627, 2621, 2999, 3533, 3631, 3697, 3989, 4057, 4133, 4523, 4603, 4733, 4919, 5273, 5591, 5641, 6211, 6247, 6269, 6299, 6469, 6803, 7753, 7879, 7937, 8353, 8543, 8971, 8999, 9041, 9181, 9413, 9479, 9787, 9887, 9941, 10487

Offset: 1

Zak Seidov and Robert Israel, Mar 18 2025

nonn

Primes in A133796.

Corresponding m's are in A092108.

			a(3) = 71 is a term because 71 = 37 + 34 is prime, where 37 is the 12th prime and 34 is the 12th semiprime.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A000040, A001358, A092108, A133796.

N:= 10000: # for terms where the m-th prime and m-th semiprime are <= N
P:= select(isprime, [2,seq(i,i=3..N,2)]): nP:= nops(P):
S:= NULL:
for i from 1 to nP while P[i]^2 <= N do
  jmax:= ListTools:-BinaryPlace(P, N/P[i]);
  S:= S, op(P[i..jmax] *~ P[i]);
od:
S:= sort([S]):
m:= min(nP, nops(S)):
select(isprime,P[1..m] + S[1..m]);

sp=Select[Range[3300],PrimeOmega[#]==2&];p=Prime[Range[Length[sp]]];Select[p+sp,PrimeQ] (* James C. McMahon, Mar 20 2025 *)

a(n) = A133796(A092108(n)).

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

Original entry on oeis.org

6, 9, 14, 25, 38, 55, 86, 122, 141, 158, 178, 185, 218, 262, 301, 326, 446, 466, 537, 634, 695, 723, 758, 785, 866, 878, 886, 895, 898, 921, 993, 1006, 1041, 1047, 1077, 1099, 1126, 1138, 1154, 1198, 1214, 1219, 1234, 1262, 1366, 1466, 1535, 1679, 1706, 1751, 1774, 1779, 1822, 1977, 2026, 2173

Offset: 1

Zak Seidov and Robert Israel, Mar 19 2025

nonn

Semiprimes in A133796.

Corresponding m's are A092021.

			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.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A092021, A133796, A382186.

N:= 10000: # for terms where the m-th prime and m-th semiprime are <= N
P:= select(isprime, [2, seq(i, i=3..N, 2)]): nP:= nops(P):
S:= NULL:
for i from 1 to nP while P[i]^2 <= N do
  jmax:= ListTools:-BinaryPlace(P, N/P[i]);
  S:= S, op(P[i..jmax] *~ P[i]);
od:
S:= sort([S]):
m:= min(nP, nops(S)):
select(t -> numtheory:-bigomega(t)=2, P[1..m] + S[1..m]);

a(n) = A133796(A092021(n)).

cp's OEIS Frontend

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

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