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

1,2

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)

Links

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

Example

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.

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 *)

Prog

(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
(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

Crossrefs

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

Sequence in context: A095402 A086294 A361632 * A361630 A323171 A008472

Adjacent sequences: A075857 A075858 A075859 * A075861 A075862 A075863

Keyword

nonn,look

Author

Joseph L. Pe, Oct 15 2002

Extensions

Better description from Labos Elemer, Apr 09 2003

Name clarified by Michel Marcus, Sep 02 2023

Status

approved