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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).
13
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).
Sequence in context: A351926 A345303 A345310 * A081758 A106492 A338038
KEYWORD
nonn
AUTHOR
STATUS
approved