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A002217 Starting with n, repeatedly calculate the sum of prime factors (with repetition) of the previous term, until reaching 0 or a fixed point: a(n) is the number of terms in the resulting sequence.
(Formerly M0150 N0060)
6

%I M0150 N0060

%S 2,1,1,1,1,2,1,3,3,2,1,2,1,4,4,4,1,4,1,4,3,2,1,4,3,5,4,2,1,3,1,3,5,2,

%T 3,3,1,4,5,2,1,3,1,5,2,4,1,2,5,3,5,2,1,2,5,2,3,2,1,3,1,6,2,3,5,5,1,4,

%U 6,5,1,3,1,6,2,2,5,5,1,2,3,2,1,5,3,3,4,2,1,2,5,5,3,6,5,2,1,5,2,5,1,3,1,2,5

%N Starting with n, repeatedly calculate the sum of prime factors (with repetition) of the previous term, until reaching 0 or a fixed point: a(n) is the number of terms in the resulting sequence.

%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

%H T. D. Noe and Christian N. K. Anderson, <a href="/A002217/b002217.txt">Table of n, a(n) for n = 1..10000</a> (first 1000 terms are from T. D. Noe)

%H Christian N. K. Anderson, <a href="/A002217/a002217.txt">n, the fixed point, a(n), and the trajectories</a> for n = 1..10000.

%H M. Lal, <a href="http://dx.doi.org/10.1090/S0025-5718-1969-0242765-9">Iterates of a number-theoretic function</a>, Math. Comp., 23 (1969), 181-183.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/SumofPrimeFactors.html">Sum of Prime Factors</a>

%e 20 -> 2+2+5 = 9 -> 3+3 = 6 -> 2+3 = 5, so a(20) = length of sequence {20,9,6,5} = 4.

%p with(numtheory): P:=proc(q) local a,b,j,k,n; print(2);

%p for n from 2 to q do a:=n; b:=a; k:=1; while not isprime(a) do k:=k+1;

%p a:=ifactors(a)[2]; a:=add(a[j][1]*a[j][2],j=1..nops(a)); if a=b then k:=k-1; break;

%p else b:=a; fi; od; print(k); od; end: P(10^4); # _Paolo P. Lava_, Apr 24 2015

%t spf[n_] := Times @@@ FactorInteger[n] // Total; a[1] = 2; a[n_] := Length[ FixedPointList[spf, n]] - 1; Array[a, 105] (* _Jean-Fran├žois Alcover_, Feb 09 2018 *)

%Y Cf. A001414 (sum of prime factors of n), A029908 (fixed point that is reached).

%K nonn

%O 1,1

%A _N. J. A. Sloane_.

%E More terms and better description from _Reinhard Zumkeller_, Apr 08 2003

%E Incorrect comment removed by _Harvey P. Dale_, Aug 16 2011

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Last modified January 16 23:44 EST 2019. Contains 319206 sequences. (Running on oeis4.)