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URL: https://oeis.org/A068318

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A068318
Sum of prime factors of n-th semiprime.
14
4, 5, 6, 7, 9, 8, 10, 13, 10, 15, 14, 19, 12, 21, 16, 25, 14, 20, 16, 22, 31, 33, 18, 26, 39, 18, 43, 22, 45, 32, 20, 34, 49, 24, 55, 40, 28, 61, 24, 22, 63, 44, 46, 26, 69, 50, 73, 24, 34, 75, 36, 81, 56, 30, 85, 26, 62, 91, 64, 42, 28, 99, 70, 103, 36, 46, 105, 30, 74, 109
OFFSET
1,1
COMMENTS
Odd k is a term if and only if k - 2 is prime. Goldbach's conjecture implies that every even number k >= 4 is a term. - Jianing Song, May 26 2021
LINKS
Robert G. Wilson v, Graph of n and a(n).
FORMULA
a(n) = A001414(A001358(n)).
a(n) = A003415(A001358(n)), the arithmetic derivative.
If A001358(n) = s*p, then in this sequence a(n) = s+p.
a(n) = A084126(n)+A084127(n). - Reinhard Zumkeller, Jul 24 2006 [Typo in formula fixed by Zak Seidov, Aug 23 2014]
EXAMPLE
a(2) = 5 because A001358(2) = 6 = 2*3 and 2+3 = 5.
MAPLE
with(numtheory): a:=proc(n) if bigomega(n)=2 and nops(factorset(n))=2 then factorset(n)[1]+factorset(n)[2] elif bigomega(n)=2 then 2*sqrt(n) else fi end: seq(a(n), n=1..214); # Emeric Deutsch
MATHEMATICA
f[n_] := Total[#1*#2 & @@@ FactorInteger@ n]; f@# & /@ Select[Range@300, PrimeOmega@# == 2 &] (* Robert G. Wilson v, Jan 23 2013 *)
PROG
(Python)
from math import isqrt
from sympy import primepi, primerange, factorint
def A068318(n):
def bisection(f, kmin=0, kmax=1):
while f(kmax) > kmax: kmax <<= 1
kmin = kmax >> 1
while kmax-kmin > 1:
kmid = kmax+kmin>>1
if f(kmid) <= kmid:
kmax = kmid
else:
kmin = kmid
return kmax
def f(x): return int(n+x+((t:=primepi(s:=isqrt(x)))*(t-1)>>1)-sum(primepi(x//p) for p in primerange(s+1)))
return sum(p*e for p, e in factorint(bisection(f, n, n)).items()) # Chai Wah Wu, Apr 03 2025
(PARI) s(n) = my(f = factor(n)); if(bigomega(f) == 2, f[, 1]~*f[, 2], 0);
list(lim) = select(x -> x > 0, apply(s, vector(lim, i, i))); \\ Amiram Eldar, May 15 2025
CROSSREFS
Semiprimes are in A001358.
Sequence in context: A143789 A068521 A196697 * A347932 A242337 A201739
KEYWORD
nonn
AUTHOR
Reinhard Zumkeller, Feb 27 2002
STATUS
approved