A016837 -id:A016837 - cp's OEIS frontend

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

Enoch Haga, Carlos Rivera, Patrick De Geest

			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.

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

Cf. A016837, A050703, A075254.

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

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 *)

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;}

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

Corrected by Michel Marcus, Jul 24 2015

A282795 Start with n. If n is 1 or a prime, stop. Otherwise, add the prime factors of n (with repetition) to n, and repeat until reaching a prime, when we stop. If no prime is ever reached, a(n) = -1.

Original entry on oeis.org

1, 2, 3, 23, 5, 11, 7, 23, 23, 17, 11, 19, 13, 23, 23, 47, 17, 41, 19, 29, 31, 47, 23, 47, 47, 41, 71, 71, 29, 71, 31, 83, 47, 53, 47, 71, 37, 59, 71, 71, 41, 83, 43, 59, 167, 71, 47, 59, 149, 167, 71, 167, 53, 83, 71, 167, 79, 89, 59, 251, 61

Offset: 1

Peter Weiss, Feb 21 2017

nonn

a(n) > 0 for every n up to at least 2000000. - Ivan N. Ianakiev, Feb 23 2017

			4 -> 4+2+2 = 8 -> 8+2+2+2 = 14 -> 14+2+7 = 23, prime, so a(4) = 23.

R. J. Mathar, Table of n, a(n) for n = 1..100000

Cf. A016837, A037274 (home primes), A075254, A195264.

A282795 := proc(n)
    local nrec;
    nrec := n ;
    while not isprime(nrec) and nrec <> 1 do
        nrec := A075254(nrec) ;
    end do:
    nrec ;
end proc:
seq(A282795(n),n=1..90) ; # R. J. Mathar, Feb 23 2017

f[n_]:=n+Total[Replace[{a_,b_}->a*b]/@FactorInteger[n]];
a[1]=1;a[n_]:=If[PrimeQ[n],n,Last[NestWhileList[f,n,!PrimeQ[#]&]]];
Array[a,2000000] (* Ivan N. Ianakiev, Feb 23 2017 *)

cp's OEIS Frontend

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.

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