OFFSET
1,2
COMMENTS
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. - 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
The asymptotic density of this sequence is Sum_{k in A001694, A000005(k) in A028982} f(k) = 0.35787373..., where f(k) = (3/(Pi^2*k)) * Product_{prime p|k} (p/(p+1)). - Amiram Eldar, Oct 07 2025
REFERENCES
Simon Colton, Automated Theorem Discovery: A Future Direction for Theorem Provers, 2002.
LINKS
Amiram Eldar, Table of n, a(n) for n = 1..10000
Simon Colton, Refactorable Numbers - A Machine Invention, J. Integer Sequences, Vol. 2 (1999), Article 99.1.2.
Simon Colton, HR - Automatic Theory Formation in Pure Mathematics, 1998-1999. [Wayback Machine link]
Simon Colton, Alan Bundy, and Toby Walsh, Automatic Concept Formation in Pure Mathematics, Proceedings of the 16th international joint conference on Artificial Intelligence - IJCAI '99, 1999; alternative link.
Simon Colton, Alan Bundy, and Toby Walsh, Agent Based Cooperative Theory Formation in Pure Mathematics, in Proceedings of AISB 2000 symposium on creative and cultural aspects and applications of AI and cognitive science, 2000, pp. 11-18.
Simon Colton, Roy McCasland, Alan Bundy, and Toby Walsh, Automated Theory Formation for Tutoring Tasks in Pure Mathematics, in Proceedings of the CADE'02 Workshop on the Role of Automated Deduction in Mathematics, 2002. [Wayback Machine link]
EXAMPLE
tau(6) = 4, which is a square number, so 6 is in this sequence.
MATHEMATICA
Select[Range[200], IntegerQ[Sqrt[DivisorSigma[0, #]]]&] (* Harvey P. Dale, Apr 20 2011 *)
PROG
(PARI) is(n)=issquare(numdiv(n)) \\ Charles R Greathouse IV, Jan 22 2013
CROSSREFS
KEYWORD
easy,nonn
AUTHOR
Simon Colton (simonco(AT)cs.york.ac.uk)
EXTENSIONS
Links corrected and edited by Daniel Forgues, Jun 30 2010 and Amiram Eldar, Oct 07 2025
STATUS
approved