A075860 a(n) is the fixed point reached when the map x -> A008472(x) is iterated, starting from x = n, with the convention a(1)=0.

0, 2, 3, 2, 5, 5, 7, 2, 3, 7, 11, 5, 13, 3, 2, 2, 17, 5, 19, 7, 7, 13, 23, 5, 5, 2, 3, 3, 29, 7, 31, 2, 3, 19, 5, 5, 37, 7, 2, 7, 41, 5, 43, 13, 2, 5, 47, 5, 7, 7, 7, 2, 53, 5, 2, 3, 13, 31, 59, 7, 61, 3, 7, 2, 5, 2, 67, 19, 2, 3, 71, 5, 73, 2, 2, 7, 5, 5, 79, 7, 3, 43, 83, 5, 13, 2, 2, 13, 89

Offset: 1

Author

Joseph L. Pe, Oct 15 2002

Keywords

• nonn
• look

Comments

For n>1, the sequence reaches a fixed point, which is prime.

From Robert Israel, Mar 31 2020: (Start)

a(n) = n if n is prime.

a(n) = n/2 + 2 if n is in A108605.

a(n) = n/4 + 2 if n is in 4*A001359. (End)

Examples

			Starting with 60 = 2^2 * 3 * 5 as the first term, add the prime factors of 60 to get the second term = 2 + 3 + 5 = 10. Then add the prime factors of 10 = 2 * 5 to get the third term = 2 + 5 = 7, which is prime. (Successive terms of the sequence will be equal to 7.) Hence a(60) = 7.
		

Links

• Robert Israel, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A008472 (sum of prime divisors of n), A029908.

Programs

Maple

f:= proc(n) option remember;
  if isprime(n) then n
  else procname(convert(numtheory:-factorset(n), `+`))
  fi
end proc:
f(1):= 0:
map(f, [$1..100]); # Robert Israel, Mar 31 2020

Mathematica

f[n_] := Module[{a}, a = n; While[ !PrimeQ[a], a = Apply[Plus, Transpose[FactorInteger[a]][[1]]]]; a]; Table[f[i], {i, 2, 100}]
(* Second program: *)
a[n_] := If[n == 1, 0, FixedPoint[Total[FactorInteger[#][[All, 1]]]&, n]];
Array[a, 100] (* Jean-François Alcover, Apr 01 2020 *)

PARI

fp(n, pn) = if (n == pn, n, fp(vecsum(factor(n)[, 1]), n));
a(n) = if (n==1, 0, fp(n, 0)); \\ Michel Marcus, Sep 02 2023

Python

from sympy import primefactors
def a(n, pn):
    if n == pn:
        return n
    else:
        return a(sum(primefactors(n)), n)
print([a(i, None) for i in range(1, 100)]) # Gleb Ivanov, Nov 05 2021

Extensions

Better description from Labos Elemer, Apr 09 2003

Name clarified by Michel Marcus, Sep 02 2023