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

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

Offset: 1

Author

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

Keywords

• nonn

Comments

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

Examples

			5040 is the smallest number with 60 divisors.
		

References

• 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.]

Links

• 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).

Crossrefs

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

Cf. A000005, A005179, A037019, A251483.

Programs

Mathematica

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 *)

Formula

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

Extensions

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