A036455 -id:A036455 - cp's OEIS frontend

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

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

Offset: 1

Labos Elemer

nonn

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

			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.

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.

Equals A060937 - 1. Cf. A007624, A036450, A036452, A036453, A036455, A030630.

Table[ Length[ FixedPointList[ DivisorSigma[0, # ] &, n]] - 2, {n, 105}] (* Robert G. Wilson v, Mar 11 2005 *)

for(x = 1,150, for(a=0,15, if(a==0,d=x, if(d<3,print(a-1),d=numdiv(d) )) ))

a(n)=my(t);while(n>2,n=numdiv(n);t++);t \\ Charles R Greathouse IV, Apr 07 2012

a(n) = a(d(n)) + 1 if n > 2.

a(n) = 1 iff n is an odd prime.

A036436 Numbers whose number of divisors is a square.

Original entry on oeis.org

1, 6, 8, 10, 14, 15, 21, 22, 26, 27, 33, 34, 35, 36, 38, 39, 46, 51, 55, 57, 58, 62, 65, 69, 74, 77, 82, 85, 86, 87, 91, 93, 94, 95, 100, 106, 111, 115, 118, 119, 120, 122, 123, 125, 129, 133, 134, 141, 142, 143, 145, 146, 155, 158, 159, 161, 166, 168, 177, 178, 183

Offset: 1

Simon Colton (simonco(AT)cs.york.ac.uk)

Invented by the HR (named after Hardy and Ramanujan) concept formation program.
Numbers in this sequence but not in A036455 are 1, 1260, 1440, 1800, 1980 etc. [From R. J. Mathar, Oct 20 2008]
tau(p^(n^2-1)) = n^2 so numbers of this form are in this sequence, and because tau is multiplicative: if a and b are in this sequence and (a,b)=1 then a*b is also in a(n). - Enrique Pérez Herrero, Jan 22 2013
What is the density of this sequence? It contains A030229 and thus has (lower) density at least 3/Pi^2 = 0.30396...; it does not contain any members of A030059 or A060687, and hence has (upper) density at most 1 - 3/Pi^2 - 6*A179119/Pi^2 = 0.49528.... - Charles R Greathouse IV, Jan 11 2025

			tau(6)=4, which is a square number, so 6 is in this sequence.

S. Colton, Automated Theorem Discovery: A Future Direction for Theorem Provers, 2002.

Amiram Eldar, Table of n, a(n) for n = 1..10000
S. Colton, Refactorable Numbers - A Machine Invention, J. Integer Sequences, Vol. 2, 1999, #2.
S. Colton, HR - Automatic Theory Formation in Pure Mathematics (Unfortunately [403 Forbidden])
S. Colton and A. Bundy, Automatic Concept Formation in Pure Mathematics
S. Colton, A. Bundy and T. Walsh, Agent Based Cooperative Theory Formation in Pure Mathematics
S. Colton, R. McCasland and A. Bundy, Automated Theory Formation for Tutoring Tasks in Pure Mathematics, 2002.

Contains A030229 as a subsequence.

Cf. A009087, A033950, A057265, A049439.

Select[Range[200],IntegerQ[Sqrt[DivisorSigma[0,#]]]&]  (* Harvey P. Dale, Apr 20 2011 *)

is(n)=issquare(numdiv(n)) \\ Charles R Greathouse IV, Jan 22 2013

Links corrected and edited by Daniel Forgues, Jun 30 2010

A036457 Numbers k for which exactly 5 applications of A000005 are needed to reach 2.

Original entry on oeis.org

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

Offset: 1

Labos Elemer

nonn

Subsequences include A030630 (numbers with 12 divisors), A030636 (numbers with 18 divisors), A030638 (numbers with 20 divisors), A137491 (numbers with 28 divisors), etc. [edited by Jon E. Schoenfield, May 12 2018]

			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.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A000005, A007624, A036450, A036452, A036453, A036455, A030630.

A036459:= proc(n) option remember;
  if n <= 2 then 0 else 1 + procname(numtheory:-tau(n)) fi
end proc:
select(A036459 = 5, [$1..1000]); # Robert Israel, Jan 25 2016

Select[Range@ 480, Last@ # == 2 && #[[5]] != 2 &@ NestList[DivisorSigma[0, #] &, #, 5] &] (* Michael De Vlieger, Jan 26 2016 *)

is(n)=for(i=1,4,n=numdiv(n); if(n<3, return(0))); numdiv(n)==2 \\ Charles R Greathouse IV, Sep 17 2015

d(d(d(d(d(a(n)))))) = 2 for all n.

A036459(a(n)) = 5. - Ivan Neretin, Jan 25 2016

New name from Robert Israel, Jan 25 2016

A053471 a(n) is the cototient of n (A051953) iterated 3 times.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 0, 1, 0, 2, 0, 2, 0, 2, 0, 2, 0, 4, 0, 4, 1, 4, 0, 4, 0, 4, 1, 4, 0, 8, 0, 4, 0, 8, 0, 8, 0, 8, 1, 8, 0, 12, 0, 8, 3, 8, 0, 8, 0, 12, 0, 8, 0, 16, 1, 8, 3, 12, 0, 16, 0, 8, 3, 8, 0, 16, 0, 16, 1, 16, 0, 16, 0, 12, 1, 16, 0, 24, 0, 16, 3, 22, 0, 24, 3, 16, 0, 16, 0, 24, 0, 16, 1, 16, 0

Offset: 1

Labos Elemer, Jan 14 2000

nonn

Iteration of A051953 behaves similarly to that of Euler Phi. Analogous 3rd iterates for A000005 or A000010 are A036455 and A049099.

It is assumed here that the value of A051953 at 0 is 0. - Antti Karttunen, Dec 22 2017

			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 so the 50th term is 12.

Antti Karttunen, Table of n, a(n) for n = 1..16384

Cf. A051953, A000005, A000010, A036455, A049099.

Cf. A053470, A053472, A053473.

Array[Nest[# - EulerPhi@ # &, #, 3] &, 95] (* Michael De Vlieger, Dec 23 2017 *)

A051953(n) = if(!n,n,(n-eulerphi(n))); \\ With modification that returns zero for zero.
A053471(n) = A051953(A051953(A051953(n))); \\ Antti Karttunen, Dec 22 2017

A036456 Numbers k for which exactly 4 applications of A000005 are needed to reach 2.

Original entry on oeis.org

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

Offset: 1

Labos Elemer

nonn

Similar to but different from A007624. Terms like 60, 72, 84, 90, 96, 108, 126, etc. are not present here.

			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.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A000005, A007624, A036450, A036452, A036453, A036455.

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

With d(n) = number of divisors(n), d(d(d(d(a(n))))) = 2 and d(d(d(a(n)))) > 2.

A036459(a(n)) = 4. - Ivan Neretin, Jan 25 2016

New name (using new name for A036457 from Robert Israel) from Jon E. Schoenfield, May 12 2018

A053477 Sum of iterates of divisor number function A000005.

Original entry on oeis.org

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

Offset: 1

Labos Elemer, Jan 14 2000

nonn

			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.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A000005, A036452, A036453, A036454, A036455, A036459.

f:= proc(n) option remember;
  if n <= 2 then n
  else n + procname(numtheory:-tau(n));
  fi
end proc:
map(f, [$1..80]); # Robert Israel, Nov 14 2016

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 *)

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

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A036436 Numbers whose number of divisors is a square.

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A036457 Numbers k for which exactly 5 applications of A000005 are needed to reach 2.

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A053471 a(n) is the cototient of n (A051953) iterated 3 times.

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A036456 Numbers k for which exactly 4 applications of A000005 are needed to reach 2.

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A053477 Sum of iterates of divisor number function A000005.

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