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Number of iterations required to reach stationary value when repeatedly applying d, the number of divisors function ( A000005).
+10
26
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
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
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) )) ))
a(n) = d(d(d(n))), the 3rd iterate of the number-of-divisors function with an initial value of n.
+10
15
1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 2, 2, 2, 2, 2, 3, 2, 3, 2, 2, 2, 3, 2, 2, 2, 3, 2, 3, 2, 3, 2, 2, 2, 2, 2, 2, 2, 3, 2, 3, 2, 3, 3, 2, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 2, 2, 4, 2, 2, 3, 2, 2, 3, 2, 3, 2, 3, 2, 4, 2, 2, 3, 3, 2, 3, 2, 3, 2, 2, 2, 4, 2, 2, 2, 3, 2, 4, 2, 3, 2, 2, 2, 4, 2, 3, 3, 2, 2, 3, 2, 3, 3
COMMENTS
The iterated d function rapidly converges to the fixed point 2.
The fourth iterate begins as follows:
1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 2, 2, 2, 2, 2, 3, 2, 2, 2, 2, 2, 3, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, ... . (End)
REFERENCES
S. Ramanujan, Collected Papers, Ed. G. H. Hardy et al., Cambridge 1927; Chelsea, NY, 1962, p. 128. - N. J. A. Sloane, Jun 02 2014
EXAMPLE
n = 5040, d(5040) = 60, d(d(5040)) = d(60) = 12 and a(5040) = d(12) = 6.
PROG
(Python)
from sympy import divisor_count
def A036450(n): return divisor_count(divisor_count(divisor_count(n))) # Chai Wah Wu, Nov 17 2022
a(n) = d(d(d(d(n)))), the 4th iterate of number-of-divisors function with initial value of n.
+10
11
1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 2, 2, 2, 2, 2, 3, 2, 2, 2, 2, 2, 3, 2, 2, 2, 2, 2, 2, 2, 2, 2
COMMENTS
The iterated d function rapidly converges to fixed point 2. For k=4, the first n for which a(n)>2 is 60.
EXAMPLE
E.g., n=96 and its successive iterates are 12,6,4,3 and 2. The 4th term is a(96)=3.
Numbers k for which exactly 5 applications of A000005 are needed to reach 2.
+10
4
60, 72, 84, 90, 96, 108, 126, 132, 140, 150, 156, 160, 180, 198, 200, 204, 220, 224, 228, 234, 240, 252, 260, 276, 288, 294, 300, 306, 308, 315, 336, 340, 342, 348, 350, 352, 360, 364, 372, 380, 392, 396, 414, 416, 420, 432, 444, 450, 460, 468, 476, 480
FORMULA
d(d(d(d(d(a(n))))))) = 2 for all n.
EXAMPLE
a(13)=180; the successive iterates are 18, 6, 4, 3, and finally the 5th is 2;
a(3)=84; divisor numbers are 12, 6, 4, 3, and 2.
MAPLE
if n <= 2 then 0 else 1 + procname(numtheory:-tau(n)) fi
end proc:
MATHEMATICA
Select[Range@ 480, Last@ # == 2 && #[[5]] != 2 &@ NestList[DivisorSigma[0, #] &, #, 5] &] (* Michael De Vlieger, Jan 26 2016 *)
a(n) is the cototient of n ( A051953) iterated 5 times.
+10
4
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 2, 0, 1, 0, 2, 0, 2, 0, 2, 0, 2, 0, 4, 0, 2, 0, 2, 0, 2, 0, 4, 0, 2, 0, 4, 0, 2, 0, 4, 0, 4, 0, 2, 0, 2, 0, 4, 0, 4, 0, 4, 0, 4, 0, 4, 0, 4, 0, 8, 0, 4, 0, 8, 0, 8, 0, 4, 0, 4, 0, 8, 0, 4, 0, 4, 0, 4, 0, 4, 0, 8, 0, 8, 0, 4, 1
EXAMPLE
n=50, n_1 = n - phi(n) = 50 - 20 = 30, n_2 = n_1 - Phi(n_1) = 30 - 8 = 22, n_3 = 22 - Phi(22) = 12, n_4 = n_3 - Phi(n_3) = 12 - 4 = 8, n_5 = 8 - Phi(8) = 4 so the 50th term is 4.
MATHEMATICA
a[n_] := Nest[# - EulerPhi[#]&, n, 5];
PROG
(PARI)
A051953(n) = if(!n, n, (n-eulerphi(n))); \\ With modification that returns zero for zero.
Numbers k for which exactly 4 applications of A000005 are needed to reach 2.
+10
2
12, 18, 20, 24, 28, 30, 32, 40, 42, 44, 45, 48, 50, 52, 54, 56, 63, 66, 68, 70, 75, 76, 78, 80, 88, 92, 98, 99, 102, 104, 105, 110, 112, 114, 116, 117, 124, 128, 130, 135, 136, 138, 144, 147, 148, 152, 153, 154, 162, 164, 165, 170, 171, 172, 174, 175, 176, 182
COMMENTS
Similar to but different from A007624. Terms like 60, 72, 84, 90, 96, 108, 126, etc. are not present here.
FORMULA
With d(n) = number of divisors(n), d(d(d(d(a(n))))) = 2 and d(d(d(a(n)))) > 2.
EXAMPLE
a(3)=20 and a(17)=63; for both x=20 and 63, d(x)=6 and d(d(x))=4, the 3rd iterates are 3 and the equilibrium value, i.e., 2 appears as 4th iterates.
PROG
(PARI) isok(n) = ((nd=numdiv(n)) != 2) && ((nd=numdiv(nd)) != 2) && ((nd=numdiv(nd)) != 2) && ((nd=numdiv(nd)) == 2); \\ Michel Marcus, Dec 30 2013 & Jan 26 2015
Sum of iterates of divisor number function A000005.
+10
2
1, 2, 5, 9, 7, 15, 9, 17, 14, 19, 13, 27, 15, 23, 24, 23, 19, 33, 21, 35, 30, 31, 25, 41, 30, 35, 36, 43, 31, 47, 33, 47, 42, 43, 44, 50, 39, 47, 48, 57, 43, 59, 45, 59, 60, 55, 49, 67, 54, 65, 60, 67, 55, 71, 64, 73, 66, 67, 61, 87, 63, 71, 78, 73, 74, 83, 69, 83, 78, 87, 73
EXAMPLE
If n is prime then the iteration sequence is {p,2} and the sum is p+2. If n=30, then iterations of the d function are {30,8,4,3,2} and their sum is a(30)=47.
MAPLE
f:= proc(n) option remember;
if n <= 2 then n
else n + procname(numtheory:-tau(n));
fi
end proc:
MATHEMATICA
g[n_] := DivisorSigma[0, n]; f[n_] := Plus @@ Drop[ FixedPointList[g, n], -1]; Table[ f[n], {n, 71}] (* Robert G. Wilson v, Dec 16 2004 *)
Infinitely refactorable numbers: numbers k such that each iteration under the map x -> A000005(x) produces a divisor of k.
+10
2
1, 2, 12, 24, 36, 60, 72, 84, 96, 108, 132, 156, 180, 204, 228, 240, 252, 276, 288, 348, 360, 372, 396, 444, 468, 480, 492, 504, 516, 564, 600, 612, 636, 640, 672, 684, 708, 720, 732, 792, 804, 828, 852, 864, 876, 936, 948, 972, 996, 1044, 1056, 1068, 1116, 1152
COMMENTS
In other words, let d^1(n) = A000005(n) and, for all positive integers k, let d^(k+1)(n) = A000005(d^k(n)). Sequence lists numbers n with the property that every such value of d^k(n) divides n. Not a subsequence of A141551: 504 is the smallest term in this sequence not member of A141551. a(n) is even for all n, since for any n >= 2, d^k(n) = 2 for some k. Proof: {d^k(n)} is a nonincreasing sequence of k, so it must stablize at a fixed point of the map x -> A000005(x), namely x = 1 or 2. But d^k(n) = 1 for some k implies that n = 1. - Jianing Song, Apr 20 2022
EXAMPLE
9 has 3 divisors, and 9 is a multiple of 3. But 3 has 2 divisors, and 9 is not a multiple of 2. Hence, 9 does not belong to this sequence.
36 has 9 divisors, 9 has 3 divisors, 3 has 2 divisors, and 9, 3, and 2 are all divisors of 36. (Since 2 has 2 divisors, all further steps produce a value of 2.) Hence, 36 belongs to this sequence.
PROG
(PARI) is_ A174457(n, d=n)=!until(d<3, n%(d=numdiv(d)) && return) \\ M. F. Hasler, Dec 05 2010, updated PARI syntax Apr 16 2022
CROSSREFS
Cf. A036459 (number of steps of the map), A000005 (d(n): number of divisors). Subsequence of A033950 (refactorable numbers: d(n) | n) and A141113 (d(d(n))| n).
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