A036436 Numbers whose number of divisors is a square.

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

Author

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

Keywords

• easy
• nonn

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. [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

Examples

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

References

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

Links

• 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.

Crossrefs

Contains A030229 as a subsequence.

Cf. A009087, A033950, A057265, A049439.

Programs

Mathematica

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

PARI

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

Extensions

Links corrected and edited by Daniel Forgues, Jun 30 2010