A321944 Starting from n, repeatedly compute the sum of the prime divisors until a fixed point or 0 is reached; a(n) is the number of terms, including n.

2, 1, 1, 2, 1, 2, 1, 2, 2, 2, 1, 2, 1, 3, 3, 2, 1, 2, 1, 2, 3, 2, 1, 2, 2, 4, 2, 3, 1, 3, 1, 2, 4, 2, 3, 2, 1, 4, 3, 2, 1, 3, 1, 2, 3, 3, 1, 2, 2, 2, 3, 4, 1, 2, 3, 3, 3, 2, 1, 3, 1, 5, 3, 2, 3, 3, 1, 2, 5, 4, 1, 2, 1, 4, 3, 4, 3, 3, 1, 2, 2, 2, 1, 3, 3, 4, 3, 2, 1, 3

Offset: 1

Author

Wilmer Emiro Castrillon Calderon, Dec 12 2018

Keywords

• nonn

Comments

a(n) is 1 + the number of iterations of n -> A008472(n) until n = A008472(n) or n=0.

The fixed points are in A075860.

For n>1 the fixed point is a prime number.

Examples

			For n=21: 21->{3,7} 3+7=10, 10->{2,5} 2+5=7, 7->{7} 7; 3 terms found {21,10,7}, therefore a(21) = 3.
For n=2: 2->{2} 2, 1 term found {2}, therefore a(2) = 1.
For n=1: 1->{} 0, 2 term found {1,0}, therefore a(1) = 2.
		

Links

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

Crossrefs

Cf. A008472, A075860, A002217.

Programs

Maple

f:= proc(n) option remember;
  if isprime(n) then 1
  else 1+procname(convert(numtheory:-factorset(n),`+`))
  fi
end proc:
f(1):= 2:
map(f, [$1..100]); # Robert Israel, Mar 30 2020

Mathematica

s[n_] := DivisorSum[n, # &, PrimeQ[#] &]; a[1] = 2; a[n_] := Length[ FixedPointList[s, n]] - 1; Array[a, 60, 0] (* Amiram Eldar, Dec 12 2018 *)

PARI

a(n)={my(k=1); while(n&&!isprime(n), k++; n=vecsum(factor(n)[, 1])); k} \\ Andrew Howroyd, Dec 12 2018