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A029908 Starting with n, repeatedly sum prime factors (with multiplicity) until reaching 0 or a fixed point. 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 (list; graph; refs; listen; history; text; internal format)
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]

LINKS

T. D. Noe and 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 *)

CROSSREFS

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

Cf. A081758, A002217, A075860.

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

Sequence in context: A086993 A238714 A289311 * A081758 A106492 A112264

Adjacent sequences:  A029905 A029906 A029907 * A029909 A029910 A029911

KEYWORD

nonn

AUTHOR

Dann Toliver

STATUS

approved

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Last modified January 20 02:13 EST 2019. Contains 319320 sequences. (Running on oeis4.)