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A036459 Number of iterations required to reach stationary value when repeatedly applying d, the number of divisors function (A000005). 25
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, 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, 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 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,4
COMMENTS
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.
Each value n > 0 occurs an infinite number of times. For positions of first occurrences of n, see A251483. - Ivan Neretin, Mar 29 2015
LINKS
Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
B. L. Mayer and L. H. A. Monteiro, On the divisors of natural and happy numbers: a study based on entropy and graphs, AIMS Mathematics (2023) Vol. 8, Issue 6, 13411-13424.
FORMULA
a(n) = a(d(n)) + 1 if n > 2.
a(n) = 1 iff n is an odd prime.
EXAMPLE
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.
MATHEMATICA
Table[ Length[ FixedPointList[ DivisorSigma[0, # ] &, n]] - 2, {n, 105}] (* Robert G. Wilson v, Mar 11 2005 *)
PROG
(PARI) for(x = 1, 150, for(a=0, 15, if(a==0, d=x, if(d<3, print(a-1), d=numdiv(d) )) ))
(PARI) a(n)=my(t); while(n>2, n=numdiv(n); t++); t \\ Charles R Greathouse IV, Apr 07 2012
CROSSREFS
Sequence in context: A191350 A329616 A304795 * A356159 A294926 A079167
KEYWORD
nonn
AUTHOR
STATUS
approved

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