A373630 a(n) is the n-th semiprime after the n-th prime.

4, 6, 10, 15, 25, 26, 35, 38, 49, 57, 58, 74, 85, 86, 91, 95, 118, 119, 123, 133, 134, 143, 146, 161, 183, 185, 187, 201, 202, 205, 218, 221, 237, 247, 265, 267, 278, 295, 299, 302, 309, 314, 326, 327, 334, 335, 362, 393, 395, 398, 403, 413, 415, 427, 446, 453, 466, 469, 473, 481, 482, 497, 519

Offset: 1

Author

Zak Seidov and Robert Israel, Jun 11 2024

Keywords

• nonn

Examples

			a(5) = 25 because the 5th prime is 11 and the first 5 semiprimes > 11 are 14,15,21,22,25.
		

Links

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

Crossrefs

Cf. A000040, A001358, A072000.

Programs

Maple

N:= 10^4: # for terms <= N
P:= select(isprime,[2,seq(i,i=3..N,2)]):
S:= select(t -> numtheory:-bigomega(t)=2, [$1..N]): nS:= nops(S):
f:= proc(n) local j;
  j:= ListTools:-BinaryPlace(S,P[n]);
  if j + n <= nS then S[j+n] else fail fi
end proc:
R:= NULL:
for i from 1 do
  v:= f(i);
  if v = fail then break fi;
  R:= R,v
od:
R;

Mathematica

seq={};Do[i=Prime[n]+1;cnt=0;While[cntJames C. McMahon, Jun 15 2024 *)

Python

from math import isqrt
from sympy import primepi, prime
def A373630(n):
    p = prime(n)
    q = n+int(sum(primepi(p//prime(k))-k+1 for k in range(1,primepi(isqrt(p))+1)))
    def f(x): return int(q+x-sum(primepi(x//prime(k))-k+1 for k in range(1, primepi(isqrt(x))+1)))
    m, k = q, f(q)
    while m != k:
        m, k = k, f(k)
    return m # Chai Wah Wu, Jul 23 2024

Formula

a(n) = A001358(n + A072000(A000040(n))).