A228578 Sum of the distinct prime factors of the squarefree semiprimes (A006881).

5, 7, 9, 8, 10, 13, 15, 14, 19, 12, 21, 16, 25, 20, 16, 22, 31, 33, 18, 26, 39, 18, 43, 22, 45, 32, 20, 34, 49, 24, 55, 40, 28, 61, 24, 63, 44, 46, 26, 69, 50, 73, 24, 34, 75, 36, 81, 56, 30, 85, 62, 91, 64, 42, 28, 99, 70, 103, 36, 46, 105, 30, 74, 109, 48, 38, 111

Offset: 1

Author

Wesley Ivan Hurt, Aug 28 2013

Keywords

• nonn
• easy

Comments

Sum of the distinct prime factors of A006881(n). If A006881(n) is even then a(n) = A006881(n)/2 + 2. If A006881(n) is odd then a(n) is even.

Examples

			a(1) = 5, since 6 is the first squarefree semiprime and the sum of the distinct prime factors of 6 is 2 + 3 = 5. a(2) = 7 since 10 is the second squarefree semiprime and the sum of the distinct prime factors of 10 is 2 + 5 = 7.
		

Links

• Michael De Vlieger, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A006881, A001414, A008472.

Programs

Mathematica

Total[First /@ FactorInteger@ #] & /@ Select[Range@ 240, PrimeNu@ # == 2 && SquareFreeQ@ # &] (* Michael De Vlieger, Oct 28 2015 *)

PARI

do(x)=my(v=List()); forprime(p=3,x\2, forprime(q=2,min(x\p,p-1), listput(v,[p*q,p+q]))); v=vecsort(Vec(v),1); apply(u->u[2],v) \\ Charles R Greathouse IV, Nov 05 2017

Python

from math import isqrt
from sympy import primepi, primerange, primefactors
def A228578(n):
    def f(x): return int(n+x+(t:=primepi(s:=isqrt(x)))+(t*(t-1)>>1)-sum(primepi(x//k) for k in primerange(1, s+1)))
    m, k = n, f(n)
    while m != k:
        m, k = k, f(k)
    return sum(primefactors(m)) # Chai Wah Wu, Aug 16 2024

Formula

a(n) = sopf(A006881(n)) = A008472(A006881(n)).

Also, a(n) = sopfr(A006881(n)) = A001414(A006881(n)) because A006881 are squarefree. - Zak Seidov, Oct 28 2015

Extensions

a(61)-a(67) corrected by Michael De Vlieger, Oct 28 2015