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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.

%I A002217 M0150 N0060 #44 Feb 16 2025 08:32:25
%S A002217 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 A002217 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 A002217 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 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.
%D A002217 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D A002217 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A002217 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 A002217 Christian N. K. Anderson, <a href="/A002217/a002217.txt">n, the fixed point, a(n), and the trajectories</a> for n = 1..10000.
%H A002217 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 A002217 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/SumofPrimeFactors.html">Sum of Prime Factors</a>
%e A002217 20 -> 2+2+5 = 9 -> 3+3 = 6 -> 2+3 = 5, so a(20) = length of sequence {20,9,6,5} = 4.
%t A002217 sopfr[n_] := Times @@@ FactorInteger[n] // Total;
%t A002217 a[1] = 2; a[n_] := Length[ FixedPointList[sopfr, n]] - 1;
%t A002217 Array[a, 105] (* _Jean-François Alcover_, Feb 09 2018 *)
%Y A002217 Cf. A001414 (sum of prime factors of n), A029908 (fixed point that is reached).
%K A002217 nonn
%O A002217 1,1
%A A002217 _N. J. A. Sloane_
%E A002217 More terms and better description from _Reinhard Zumkeller_, Apr 08 2003
%E A002217 Incorrect comment removed by _Harvey P. Dale_, Aug 16 2011