A189394 -id:A189394 - cp's OEIS frontend

A335831 Numbers k with a record value of tau(tau(k)) (A010553), where tau(k) is the number of divisors of k (A000005).

1, 2, 6, 12, 60, 360, 1260, 2520, 5040, 55440, 277200, 720720, 3603600, 61261200, 129729600, 908107200, 2205403200, 15437822400, 293318625600, 3226504881600, 6746328388800, 74209612276800, 195643523275200, 1855240306920000, 2152078756027200, 27977023828353600

Offset: 1

Amiram Eldar, Jun 25 2020

nonn

First differs from A189394 at n=15.

The corresponding record values are 1, 2, 3, 4, 6, 8, 9, 10, 12, 16, 18, 20, 24, 30, ... (see the link for more values).

Amiram Eldar, Table of n, a(n) for n = 1..73
Yvonne Buttkewitz, Christian Elsholtz, Kevin Ford and Jan-Christoph Schlage-Puchta, A problem of Ramanujan, Erdős, and Kátai on the iterated divisor function, International Mathematics Research Notices, Vol. 2012, No. 17 (2012), pp. 4051-4061, preprint, arXiv:1108.1815 [math.NT], 2011.
Amiram Eldar, Table of n, a(n), A010553(a(n)) for n = 1..73
Christian Elsholtz, Marc Technau and Niclas Technau, The maximal order of iterated multiplicative functions, Mathematika, Vol. 65, No. 4 (2019), pp. 990-1009, preprint, arXiv:1709.04799 [math.NT], 2017 and 2019.

Subsequence of A025487.

Cf. A000005, A005179, A010553, A189394, A193987.

f[n_] := DivisorSigma[0, DivisorSigma[0, n]]; fm = 0; s = {}; Do[f1 = f[n]; If[f1 > fm, fm = f1; AppendTo[s, n]], {n, 1, 10^5}]; s

tau(tau(a(n))) ~ c * sqrt(log(a(n)))/log(log(a(n))), where c is a constant (Buttkewitz et al., 2012).

A141320 Both n and the smallest number with n divisors are in A002182.

Original entry on oeis.org

1, 2, 4, 6, 12, 24, 36, 48, 60, 120, 180, 240, 360, 720, 1680, 5040, 10080, 20160

Offset: 1

J. Lowell, Aug 02 2008

Question: is this sequence finite? See A189394 for detailed information.

Intersection of A002182 and A002183. - Jianing Song, Apr 03 2018

			Both 20160 and the smallest number with 20160 divisors, 195643523275200, are in A002182, so 20160 is a term.

Cf. A002182, A002183, A189394.

a(n) = tau(A189394(n)) = A000005(A189394(n)). - Jianing Song, Apr 03 2018

a(16)-a(18) from Jianing Song, Apr 03 2018

Keywords fini, full are copied over from A189394 by Max Alekseyev, Feb 02 2025

A333328 Irregular triangle read by rows: T(n,0) = A002182(n) and T(n,k + 1) = A000005(T(n,k)), terminating at the first number which is not highly composite, n > 2.

Original entry on oeis.org

4, 3, 6, 4, 3, 12, 6, 4, 3, 24, 8, 36, 9, 48, 10, 60, 12, 6, 4, 3, 120, 16, 180, 18, 240, 20, 360, 24, 8, 720, 30, 840, 32, 1260, 36, 9, 1680, 40, 2520, 48, 10, 5040, 60, 12, 6, 4, 3, 7560, 64, 10080, 72, 15120, 80, 20160, 84, 25200, 90, 27720, 96, 45360, 100

Offset: 3

Davis Smith, Mar 15 2020

There are two questions related to this array: First, which rows have length greater than any previous row? Second, are there any rows which terminate at a k greater than 6?

			The irregular triangle T(n,k) starts:
  n\k   0   1   2   3   4   ...
   3:   4   3
   4:   6   4   3
   5:  12   6   4   3
   6:  24   8
   7:  36   9
   8:  48  10
   9:  60  12   6   4   3
  10: 120  16
  11: 180  18
  12: 240  20
  13: 360  24   8
  ...

James Grime and Brady Haran, 5040 and other Anti-Prime Numbers, Numberphile video (2016).

Cf. A000005, A002182, A002183, A189394.

A333328_rows(n)={my(N=Map(Mat([1,1;2,2;m=4,3])),p=2,F=[]); while(#Np,mapput(N,m,p=numdiv(m)); my(M=List([m,q=p])); while(mapisdefined(N,q,&q),listput(M,q));print(#N", "Vec(M)); F=concat(F,Vec(M))); my(s=if(m>=720720,360360,m>=5040,2520,m>=840,420,m>=60,60,2)); until(numdiv(m+=s)>p,));F}

T(n,0) = A002182(n), T(n,k) = A000005(T(n,k - 1)).

cp's OEIS Frontend

A335831 Numbers k with a record value of tau(tau(k)) (A010553), where tau(k) is the number of divisors of k (A000005).

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A141320 Both n and the smallest number with n divisors are in A002182.

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A333328 Irregular triangle read by rows: T(n,0) = A002182(n) and T(n,k + 1) = A000005(T(n,k)), terminating at the first number which is not highly composite, n > 2.

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