A228578 - OEIS

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

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.

LINKS

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

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

EXAMPLE

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.

MATHEMATICA

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

PROG

(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

CROSSREFS

Cf. A006881, A001414, A008472.

Sequence in context: A135913 A308713 A109352 * A242742 A323602 A164029