A123529 Denominator of average of prime factors of n.

1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 3, 1, 2, 1, 1, 1, 3, 1, 1, 1, 2, 1, 4, 1, 2, 1, 3, 1, 3, 1, 1, 1, 2, 1, 2, 1, 2, 1, 4, 1, 1, 1, 1, 3, 2, 1, 5, 1, 1, 1, 3, 1, 4, 1, 4, 1, 2, 1, 1, 1, 2, 3, 1, 1, 3, 1, 1, 1, 3, 1, 5, 1, 2, 3, 3, 1, 1, 1, 5, 1, 2, 1, 2, 1, 2, 1, 4, 1, 4, 1, 1, 1, 2, 1, 6, 1, 3, 3, 2, 1, 3, 1, 4, 1, 2

Offset

2,5

Comments

Prime factors counted with multiplicity. - Harvey P. Dale, Jun 20 2013

Positions of 1's are A078175. a(n) is a divisor of Omega(n) = A001222(n). The average of prime indices (as opposed to prime factors) of n is A326567(n)/A326568(n). - Gus Wiseman, Jul 18 2019

Links

G. C. Greubel, Table of n, a(n) for n = 2..5000

Mathematica

Table[Denominator[Mean[Flatten[Table[#[[1]], {#[[2]]}]&/@ FactorInteger[ n]]]], {n, 110}] (* Harvey P. Dale, Jun 20 2013 *)

Prog

(Python)
from math import gcd
from sympy import factorint
def A123529(n):
    f = factorint(n)
    return (l:=sum(f.values()))//gcd(l, sum(p*e for p, e in f.items())) # Chai Wah Wu, Jun 25 2026
(PARI) a(n) = my(f=factor(n)); denominator(sum(i=1, #f~, f[i, 1]*f[i, 2])/bigomega(f)); \\ Bruce Nye, Jun 26 2026

Crossrefs

See A123528 for more formulas and references.

Cf. A078174, A078175, A112798, A316413, A326567/A326568, A326619/A326620, A326621.

Sequence in context: A206921 A386261 A327402 * A140747 A330757 A322373

Adjacent sequences: A123526 A123527 A123528 * A123530 A123531 A123532

Keyword

frac,nonn,changed

Author

Franklin T. Adams-Watters, Oct 02 2006

Status

approved