A109352 a(n) = sum of the prime divisors of the n-th squarefree composite number.

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

Offset

1,1

Comments

Similar to the definition of A120944 except we list the sum of the divisors of n instead of n.

Links

Robert Israel, Table of n, a(n) for n = 1..10000

Formula

a(n) = A008472(A120944(n)) = A001414(A120944(n)).

Example

The 3rd squarefree composite number is 14 = 2*7, so a(3) = 2 + 7 = 9.

Maple

map(t-> convert(numtheory:-factorset(t), `+`), select(numtheory:-issqrfree and not isprime, [$6..1000])); # Robert Israel, Oct 09 2015

Mathematica

lim = 200; Total@ Map[First, FactorInteger@ #] & /@ Select[Range@ lim, SquareFreeQ@ # && CompositeQ@ # &] (* Michael De Vlieger, Oct 09 2015 *)

Prog

(PARI) distinct(n) = \\ Sum of the distinct divisors p1, p2.. of n
if p1*p2..=n { local(a, x, m, p, ln, s); for(m=2, n, p=1; s=0; a=ifactord(m); ln=length(a); if(ln > 1, for(x=1, ln, p*=a[x]; s+=a[x]; ) ); if(p==m, print1(s", ") ) ) }
ifactord(n, m=0) = \\The vector of the distinct integer factors of n.
{ local(f, j, k, flist); flist=[]; f=Vec(factor(n, m)); for(j=1, length(f[1]), flist = concat(flist, f[1][j]) ); return(flist) }
(Python)
from math import isqrt
from sympy import primepi, mobius
def A109352(n):
    def f(x): return n+1+primepi(x)+x-sum(mobius(k)*(x//k**2) for k in range(1, isqrt(x)+1))
    m, k = n+1, f(n+1)
    while m != k:
        m, k = k, f(k)
    return sum(primefactors(m)) # Chai Wah Wu, Aug 02 2024

Crossrefs

Cf. A000469, A001414, A008472, A120944.

Sequence in context: A177705 A135913 A308713 * A228578 A242742 A323602

Adjacent sequences: A109349 A109350 A109351 * A109353 A109354 A109355

Keyword

easy,nonn

Author

Cino Hilliard, Aug 21 2005

Status

approved