cp's OEIS Frontend

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

%I A228578 #48 Aug 26 2024 07:40:45
%S A228578 5,7,9,8,10,13,15,14,19,12,21,16,25,20,16,22,31,33,18,26,39,18,43,22,
%T A228578 45,32,20,34,49,24,55,40,28,61,24,63,44,46,26,69,50,73,24,34,75,36,81,
%U A228578 56,30,85,62,91,64,42,28,99,70,103,36,46,105,30,74,109,48,38,111
%N A228578 Sum of the distinct prime factors of the squarefree semiprimes (A006881).
%C A228578 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.
%H A228578 Michael De Vlieger, <a href="/A228578/b228578.txt">Table of n, a(n) for n = 1..10000</a>
%F A228578 a(n) = sopf(A006881(n)) = A008472(A006881(n)).
%F A228578 Also, a(n) = sopfr(A006881(n)) = A001414(A006881(n)) because A006881 are squarefree. - _Zak Seidov_, Oct 28 2015
%e A228578 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.
%t A228578 Total[First /@ FactorInteger@ #] & /@ Select[Range@ 240, PrimeNu@ # == 2 && SquareFreeQ@ # &] (* _Michael De Vlieger_, Oct 28 2015 *)
%o A228578 (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
%o A228578 (Python)
%o A228578 from math import isqrt
%o A228578 from sympy import primepi, primerange, primefactors
%o A228578 def A228578(n):
%o A228578     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)))
%o A228578     m, k = n, f(n)
%o A228578     while m != k:
%o A228578         m, k = k, f(k)
%o A228578     return sum(primefactors(m)) # _Chai Wah Wu_, Aug 16 2024
%Y A228578 Cf. A006881, A001414, A008472.
%K A228578 nonn,easy
%O A228578 1,1
%A A228578 _Wesley Ivan Hurt_, Aug 28 2013
%E A228578 a(61)-a(67) corrected by _Michael De Vlieger_, Oct 28 2015