A029908 Starting with n, repeatedly sum prime factors (with multiplicity) until reaching 0 or a fixed point. Then a(n) is the fixed point (or 0).

0, 2, 3, 4, 5, 5, 7, 5, 5, 7, 11, 7, 13, 5, 5, 5, 17, 5, 19, 5, 7, 13, 23, 5, 7, 5, 5, 11, 29, 7, 31, 7, 5, 19, 7, 7, 37, 7, 5, 11, 41, 7, 43, 5, 11, 7, 47, 11, 5, 7, 5, 17, 53, 11, 5, 13, 13, 31, 59, 7, 61, 5, 13, 7, 5, 5, 67, 7, 5, 5, 71, 7, 73, 5, 13, 23, 5, 5, 79, 13, 7, 43, 83, 5, 13

Offset

1,2

Comments

That is, the sopfr function (see A001414) applied repeatedly until reaching 0 or a fixed point.

For n > 1, the sequence reaches a fixed point which is either 4 or a prime.

A002217(n) is number of terms in sequence from n to a(n). - Reinhard Zumkeller, Apr 08 2003

Because sopfr(n) <= n (with equality at 4 and the primes), the first appearance of all primes is in the natural order: 2,3,5,7,11,... . - Zak Seidov, Mar 14 2011

The terms 0, 2, 3 and 4 occur exactly once, because no number > 5 can have factors that sum to be < 5, and therefore can never enter a trajectory that will drop below 5. - Christian N. K. Anderson, May 19 2013

For all primes p, where p is contained in A001359, then a(p^2) = p + 2. (A006512). Proof: p^2 has prime factors (p, p). This sums to 2p. 2p has factors (2, p). This sums to p + 2. Since p was the lesser of a twin prime, then p + 2 is the greater of a twin prime. - Ryan Bresler, Nov 04 2021

Links

Christian N. K. Anderson, Table of n, a(n) for n = 1..10000 (first 1000 terms from T. D. Noe)

Eric Weisstein's World of Mathematics, Sum of Prime Factors.

Example

20 -> 2+2+5 = 9 -> 3+3 = 6 -> 2+3 = 5, so a(20)=5.

Maple

f:= proc(n) option remember;
if isprime(n) then n
else `procname`(add(x[1]*x[2], x = ifactors(n)[2]))
fi
end proc:
f(1):= 0: f(4):= 4:
map(f, [$1..100]); # Robert Israel, Apr 27 2015

Mathematica

ffi[x_] := Flatten[FactorInteger[x]] lf[x_] := Length[FactorInteger[x]] ba[x_] := Table[Part[ffi[x], 2*w-1], {w, 1, lf[x]}] ep[x_] := Table[Part[ffi[x], 2*w], {w, 1, lf[x]}] slog[x_] := slog[x_] := Apply[Plus, ba[x]*ep[x]] Table[FixedPoint[slog, w], {w, 1, 128}]
f[n_] := Plus @@ Flatten[ Table[ #[[1]], {#[[2]]}] & /@ FactorInteger@n]; Array[ FixedPoint[f, # ] &, 87] (* Robert G. Wilson v, Jan 18 2006 *)
fz[n_]:=Plus@@(#[[1]]*#[[2]]&/@FactorInteger@n); Array[FixedPoint[fz, #]&, 1000] (* Zak Seidov, Mar 14 2011 *)

Prog

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

Crossrefs

Cf. A001414 (sum of prime factors of n).

Cf. A081758, A002217, A075860.

Cf. A001414, A056239, A008475, A082081, A082083.

Sequence in context: A351926 A345303 A345310 * A081758 A106492 A338038

Adjacent sequences: A029905 A029906 A029907 * A029909 A029910 A029911

Keyword

nonn

Author

Dann Toliver

Status

approved