A115392 First appearance of n-th prime as prime factor in list of semiprimes.

1, 2, 4, 5, 8, 10, 12, 14, 16, 21, 22, 25, 27, 29, 33, 35, 38, 41, 45, 47, 50, 52, 55, 58, 62, 64, 67, 70, 73, 76, 82, 84, 87, 88, 93, 96, 100, 104, 107, 111, 113, 115, 120, 121, 124, 126, 133, 137, 141, 142, 143, 147, 149, 155, 158, 162, 168, 169, 174, 176, 178, 183

Offset: 1

Author

Zak Seidov, Mar 08 2006

Keywords

• nonn

Comments

The positions of even semiprimes in A001358. - Juri-Stepan Gerasimov, Apr 11 2010

Examples

			a(5)=8 because 5th prime, 11, first appeared as a prime factor in 8th semiprime 22=2*11;
a(6)=10 because 6th prime, 13, first appeared as a prime factor in 10th semiprime 26=2*13.
		

Links

• Zak Seidov, Table of n, a(n) for n = 1..1000

Crossrefs

Cf. A000040, A001358.

Programs

Mathematica

sp = Select[Range[4,20000],2 == PrimeOmega[#]&]; Table[Position[Mod[sp, Prime[k]],0][[1,1]],{k,1000}] (* For first 1000 terms. - Zak Seidov, Jun 27 2017 *)

PARI

first(n) = my(l = List([4,1]), u = 2*prime(n), res = vector(n), t=0); forprime(p = 2, t++; sqrt(2*prime(n)), forprime(q = p+1, u\p, listput(l, [p*q, t]));  listsort(l); for(i=1, #l, if(res[l[i][3]]==0, res[l[i][3]] = i)) \\ David A. Corneth, Jun 28 2017

Python

from math import isqrt
from sympy import primepi, prime, primerange
def A115392(n): return int(-((t:=primepi(s:=isqrt(a:=prime(n)<<1)))*(t-1)>>1)+sum(primepi(a//p) for p in primerange(s+1))) # Chai Wah Wu, Apr 03 2025

Formula

A001358(a(n)) = 2*A000040(n) - Zak Seidov, Jun 27 2017

Extensions

Edited by Zak Seidov, Jun 27 2017