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.

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, 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, 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

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn

Examples

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

References

• N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

• T. D. Noe and Christian N. K. Anderson, Table of n, a(n) for n = 1..10000 (first 1000 terms are from T. D. Noe)
• Christian N. K. Anderson, n, the fixed point, a(n), and the trajectories for n = 1..10000.
• M. Lal, Iterates of a number-theoretic function, Math. Comp., 23 (1969), 181-183.
• Eric Weisstein's World of Mathematics, Sum of Prime Factors

Crossrefs

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

Programs

Mathematica

sopfr[n_] := Times @@@ FactorInteger[n] // Total;
a[1] = 2; a[n_] := Length[ FixedPointList[sopfr, n]] - 1;
Array[a, 105] (* Jean-François Alcover, Feb 09 2018 *)

Extensions

More terms and better description from Reinhard Zumkeller, Apr 08 2003

Incorrect comment removed by Harvey P. Dale, Aug 16 2011