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

%I A349824 #44 May 07 2025 14:57:55
%S A349824 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,
%T A349824 20,30,27,33,29,30,31,50,28,38,24,40,37,42,32,44,41,36,43,45,33,50,47,
%U A349824 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
%N A349824 a(0) = 0; for n >= 1, a(n) = (number of prime factors, counted with repetition) * (sum of prime factors, counted with repetition).
%C A349824 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).
%C A349824 Suggested by Mike Klein in an email, Dec 31 2021.
%C A349824 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).
%C A349824 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
%C A349824 From _Gary Detlefs_, Jan 14 2022: (Start)
%C A349824 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.
%C A349824 Values of n for which the trajectory terminates at 27 are {9,12,20,21,25}. (End)
%H A349824 Rémy Sigrist, <a href="/A349824/b349824.txt">Table of n, a(n) for n = 0..10000</a>
%e A349824 If n = 27 = 3^3, a(n) = 3*(3+3+3) = 27.
%e A349824 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, ...
%e A349824 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, ...
%t A349824 {0, 0}~Join~Array[Total[#]*Length[#] &@ Flatten[ConstantArray[#1, #2] & @@@ FactorInteger[#]] &, 72, 2] (* _Michael De Vlieger_, Jan 02 2022 *)
%o A349824 (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
%o A349824 (Python)
%o A349824 from sympy import factorint
%o A349824 def a(n):
%o A349824     if n == 0: return 0
%o A349824     f = factorint(n)
%o A349824     return sum(f.values()) * sum(p*e for p, e in f.items())
%o A349824 print([a(n) for n in range(74)]) # _Michael S. Branicky_, Jan 02 2022
%Y A349824 Cf. A001222, A001414.
%K A349824 nonn
%O A349824 0,3
%A A349824 _N. J. A. Sloane_, Jan 01 2022