A018845 Number of iterations required for the sum of n and its prime divisors = t to reach a prime (where t replaces n in each iteration) in A016837.

4, 2, 3, 2, 1, 2, 2, 2, 1, 3, 1, 2, 1, 1, 3, 2, 2, 2, 1, 1, 2, 2, 2, 2, 1, 3, 3, 2, 3, 5, 4, 1, 1, 1, 2, 2, 1, 2, 2, 10, 3, 2, 1, 6, 1, 3, 1, 5, 5, 1, 5, 3, 2, 1, 5, 1, 1, 2, 7, 3, 4, 4, 4, 1, 10, 3, 1, 4, 6, 3, 6, 3, 1, 6, 3, 4, 2, 2, 2, 2, 9, 2, 5, 1, 1, 3

Offset: 2

Author

Enoch Haga, Carlos Rivera, Patrick De Geest

Keywords

• easy
• nonn

Examples

			Starting with 4, 4=2*2, so 4+2+2=8. 8=2*2*2 so 8+2+2+2=14. 14=2*7 so 14+2+7=23, prime in 3 iterations, so a(4)=3.
		

Links

• Robert Israel, Table of n, a(n) for n = 2..10000

Crossrefs

Cf. A016837, A050703, A075254.

Programs

Maple

f:= proc(n) option remember; local t;
   t:= n + convert(map(convert,ifactors(n)[2],`*`),`+`);
   if isprime(t) then 1 else 1+procname(t) fi
end proc:
map(f, [$2..100]); # Robert Israel, Jul 26 2015

Mathematica

a[n_] := a[n] = Module[{t, f = FactorInteger[n]}, t = n + f[[All, 1]]. f[[All, 2]]; If[PrimeQ[t], 1, 1 + a[t]]];
a /@ Range[2, 100] (* Jean-François Alcover, Jul 19 2020, after Maple *)

PARI

sfpn(n) = {my(f = factor(n)); n + sum(k=1, #f~, f[k,1]*f[k,2]);}
a(n) = {nb = 1; while (! isprime(t=sfpn(n)), n=t; nb++); nb;}

Formula

Factor n, add n and its prime divisors. Sum = t, t replaces n, repeat until a prime is produced in k iterations.

For x in A050703, a(x) = 1. - Michel Marcus, Jul 24 2015

Number of iterations x->A075254(x) to reach a prime, starting at x=n. - R. J. Mathar, Jul 27 2015

Extensions

Corrected by Michel Marcus, Jul 24 2015