

A349824


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


5



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
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OFFSET

0,3


COMMENTS

More precisely, a(n) = (number of prime factors of n, counted with repetition) * (sum of primes 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)


LINKS

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


EXAMPLE

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, ...


MATHEMATICA

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


PROG

(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


CROSSREFS

Cf. A001222, A001414.
Sequence in context: A114340 A126102 A011433 * A332221 A340393 A332222
Adjacent sequences: A349821 A349822 A349823 * A349825 A349826 A349827


KEYWORD

nonn


AUTHOR

N. J. A. Sloane, Jan 01 2022


STATUS

approved



