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A036459 Number of iterations required to reach stationary value when repeatedly applying d, the number of divisors function (A000005).

%I A036459 #29 Apr 11 2023 16:08:16
%S A036459 0,0,1,2,1,3,1,3,2,3,1,4,1,3,3,2,1,4,1,4,3,3,1,4,2,3,3,4,1,4,1,4,3,3,
%T A036459 3,3,1,3,3,4,1,4,1,4,4,3,1,4,2,4,3,4,1,4,3,4,3,3,1,5,1,3,4,2,3,4,1,4,
%U A036459 3,4,1,5,1,3,4,4,3,4,1,4,2,3,1,5,3,3,3,4,1,5,3,4,3,3,3,5,1,4,4
%N A036459 Number of iterations required to reach stationary value when repeatedly applying d, the number of divisors function (A000005).
%C A036459 Iterating d for n, the prestationary prime and finally the fixed value of 2 is reached in different number of steps; a(n) is the number of required iterations.
%C A036459 Each value n > 0 occurs an infinite number of times. For positions of first occurrences of n, see A251483. - _Ivan Neretin_, Mar 29 2015
%H A036459 Charles R Greathouse IV, <a href="/A036459/b036459.txt">Table of n, a(n) for n = 1..10000</a>
%H A036459 B. L. Mayer and L. H. A. Monteiro, <a href="https://doi.org/10.3934/math.2023679">On the divisors of natural and happy numbers: a study based on entropy and graphs</a>, AIMS Mathematics (2023) Vol. 8, Issue 6, 13411-13424.
%F A036459 a(n) = a(d(n)) + 1 if n > 2.
%F A036459 a(n) = 1 iff n is an odd prime.
%e A036459 If n=8, then d(8)=4, d(d(8))=3, d(d(d(8)))=2, which means that a(n)=3. In terms of the number of steps required for convergence, the distance of n from the d-equilibrium is expressed by a(n). A similar method is used in A018194.
%t A036459 Table[ Length[ FixedPointList[ DivisorSigma[0, # ] &, n]] - 2, {n, 105}] (* _Robert G. Wilson v_, Mar 11 2005 *)
%o A036459 (PARI) for(x = 1,150, for(a=0,15, if(a==0,d=x, if(d<3,print(a-1),d=numdiv(d) )) ))
%o A036459 (PARI) a(n)=my(t);while(n>2,n=numdiv(n);t++);t \\ _Charles R Greathouse IV_, Apr 07 2012
%Y A036459 Equals A060937 - 1. Cf. A007624, A036450, A036452, A036453, A036455, A030630.
%K A036459 nonn
%O A036459 1,4
%A A036459 _Labos Elemer_