A349824 a(0) = 0; for n >= 1, a(n) = (number of prime factors, counted with repetition) * (sum of prime factors, counted with repetition).

0, 0, 2, 3, 8, 5, 10, 7, 18, 12, 14, 11, 21, 13, 18, 16, 32, 17, 24, 19, 27, 20, 26, 23, 36, 20, 30, 27, 33, 29, 30, 31, 50, 28, 38, 24, 40, 37, 42, 32, 44, 41, 36, 43, 45, 33, 50, 47, 55, 28, 36, 40, 51, 53, 44, 32, 52, 44, 62, 59, 48, 61, 66, 39, 72, 36, 48, 67, 63, 52, 42, 71, 60, 73

Offset: 0

Author

N. J. A. Sloane, Jan 01 2022

Keywords

• nonn

Comments

More precisely, a(n) = (number of prime factors of n, counted with repetition) * (sum of prime factors of n, counted with repetition): a(n) = A001414(n) * A001222(n).

Suggested by Mike Klein in an email, Dec 31 2021.

Conjecture (Mike Klein): Iterating n -> a(n) eventually leads to one of the fixed points {primes union 0, 27, 30} or the loop (28, 33).

It appears that with the exception of {1,4,16,27,30}, n divides a(n) iff n is prime. - Gary Detlefs, Jan 11 2022

From Gary Detlefs, Jan 14 2022: (Start)

The loop (28,33) referenced above would more correctly be denoted by (33,28). The only value of n which reaches 28 before 33 is 49.

Values of n for which the trajectory terminates at 27 are {9,12,20,21,25}. (End)

Examples

			If n = 27 = 3^3, a(n) = 3*(3+3+3) = 27.
If we start with n = 4, iterating this map produces the trajectory 4, 8, 18, 24, 36, 40, 44, 45, 33, 28, 33, 28, 33, 28, ...
If we start with n = 6, iterating this map produces the trajectory 6, 10, 14, 18, 24, 36, 40, 44, 45, 33, 28, 33, 28, 33, 28, ...
		

Links

• Rémy Sigrist, Table of n, a(n) for n = 0..10000

Crossrefs

Cf. A001222, A001414.

Programs

Mathematica

{0, 0}~Join~Array[Total[#]*Length[#] &@ Flatten[ConstantArray[#1, #2] & @@@ FactorInteger[#]] &, 72, 2] (* Michael De Vlieger, Jan 02 2022 *)

PARI

a(n) = { if (n==0, 0, my (f=factor(n)); bigomega(f)*sum(k=1, #f~, f[k,1]*f[k,2])) } \\ Rémy Sigrist, Jan 01 2022

Python

from sympy import factorint
def a(n):
    if n == 0: return 0
    f = factorint(n)
    return sum(f.values()) * sum(p*e for p, e in f.items())
print([a(n) for n in range(74)]) # Michael S. Branicky, Jan 02 2022