A251483 -id:A251483 - 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.

A009287 a(1) = 3; thereafter a(n+1) = least k with a(n) divisors.

Original entry on oeis.org

3, 4, 6, 12, 60, 5040, 293318625600, 670059168204585168371476438927421112933837297640990904154667968000000000000

Offset: 1

David W. Wilson and James Kilfiger (jamesk(AT)maths.warwick.ac.uk)

nonn

The sequence must start with 3, since a(1)=1 or a(1)=2 would lead to a constant sequence. - M. F. Hasler, Sep 02 2008
The calculation of a(7) and a(8) is based upon the method in A037019 (which, apparently, is the method previously used by the authors of A009287). So a(7) and a(8) are correct unless n=a(6)=5040 or n=a(7)=293318625600 are "exceptional" as described in A037019. - Rick L. Shepherd, Aug 17 2006
a(7) is correct because 5040 is not exceptional (see A072066). - T. D. Noe, Sep 02 2008
Terms from a(2) to a(7) are highly composite (that is, found in A002182), but a(8) is not. - Ivan Neretin, Mar 28 2015 [Equivalently, the first 6 terms are in A002183, but a(7) is not. Note that the smallest number with at least a(7) divisors is A002182(695) ~ 1.77 * 10^59 with 293534171136 divisors, which is much smaller than a(8) ~ 6.70 * 10^75. - Jianing Song, Jul 15 2021]
Grime reported that Ramanujan unfortunately missed a(7) with 5040 divisors. - Frank Ellermann, Mar 12 2020
It is possible to prepend 2 to this sequence as follows. a(0) = 2; for n > 0, a(n) = the smallest natural number greater than a(n-1) with a(n-1) divisors. - Hal M. Switkay, Jul 03 2022

			5040 is the smallest number with 60 divisors.

Amarnath Murthy, Pouring a few more drops in the ocean of Smarandache Sequences and Conjectures (to be published in the Smarandache Notions Journal) [Note: this author submitted two erroneous versions of this sequence to the OEIS, A036460 and A061080, entries which contained invalid conjectures.]

Jason Earls, A note on the Smarandache divisors of divisors sequence and two similar sequences, in Smarandache Notions Journal (2004), Vol. 14.1, page 274.
James Grime and Brady Haran, Infinite Anti-Primes, Numberphile video (2016).

Coincides with A251483 for 1 <= n <= 7 (only).

Cf. A000005, A005179, A037019, A251483.

f[n_] := Block[{k = 3, s = (Times @@ (Prime[Range[Length@ #]]^Reverse[# - 1])) & @ Flatten[FactorInteger[#] /. {a_Integer, b_} :> Table[a, {b}]] & /@ Range@ 10000}, Reap@ Do[Sow[k = s[[k]]], {n}] // Flatten // Rest]; f@ 6 (* Michael De Vlieger, Mar 28 2015, after Wouter Meeussen at A037019 *)

a(n) = A005179(a(n-1)).

Entry revised by N. J. A. Sloane, Aug 25 2006

cp's OEIS Frontend

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

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A009287 a(1) = 3; thereafter a(n+1) = least k with a(n) divisors.

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A036460 Erroneous version of A009287 (the 2 should be omitted).

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