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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 (list; graph; refs; listen; history; text; internal format)
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

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Last modified February 1 02:31 EST 2023. Contains 359981 sequences. (Running on oeis4.)