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A036436 Numbers whose number of divisors is a square.

%I A036436 #31 Jan 11 2025 18:47:53
%S A036436 1,6,8,10,14,15,21,22,26,27,33,34,35,36,38,39,46,51,55,57,58,62,65,69,
%T A036436 74,77,82,85,86,87,91,93,94,95,100,106,111,115,118,119,120,122,123,
%U A036436 125,129,133,134,141,142,143,145,146,155,158,159,161,166,168,177,178,183
%N A036436 Numbers whose number of divisors is a square.
%C A036436 Invented by the HR (named after Hardy and Ramanujan) concept formation program.
%C A036436 Numbers in this sequence but not in A036455 are 1, 1260, 1440, 1800, 1980 etc. [From _R. J. Mathar_, Oct 20 2008]
%C A036436 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
%C A036436 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
%D A036436 S. Colton, Automated Theorem Discovery: A Future Direction for Theorem Provers, 2002.
%H A036436 Amiram Eldar, <a href="/A036436/b036436.txt">Table of n, a(n) for n = 1..10000</a>
%H A036436 S. Colton, <a href="http://www.cs.uwaterloo.ca/journals/JIS/colton/joisol.html">Refactorable Numbers - A Machine Invention</a>, J. Integer Sequences, Vol. 2, 1999, #2.
%H A036436 S. Colton, <a href="http://www.dai.ed.ac.uk/homes/simonco/research/hr/">HR - Automatic Theory Formation in Pure Mathematics</a> (Unfortunately [403 Forbidden])
%H A036436 S. Colton and A. Bundy, <a href="https://citeseerx.ist.psu.edu/pdf/dc6a2b3ca6b6e3e2f4565ff0cbc09160b29160cd">Automatic Concept Formation in Pure Mathematics</a>
%H A036436 S. Colton, A. Bundy and T. Walsh, <a href="http://www.doc.ic.ac.uk/~sgc/papers/colton_aisb00.pdf">Agent Based Cooperative Theory Formation in Pure Mathematics</a>
%H A036436 S. Colton, R. McCasland and A. Bundy, <a href="http://www.doc.ic.ac.uk/~sgc/html_papers/colton_radm02.html">Automated Theory Formation for Tutoring Tasks in Pure Mathematics</a>, 2002.
%e A036436 tau(6)=4, which is a square number, so 6 is in this sequence.
%t A036436 Select[Range[200],IntegerQ[Sqrt[DivisorSigma[0,#]]]&]  (* _Harvey P. Dale_, Apr 20 2011 *)
%o A036436 (PARI) is(n)=issquare(numdiv(n)) \\ _Charles R Greathouse IV_, Jan 22 2013
%Y A036436 Contains A030229 as a subsequence.
%Y A036436 Cf. A009087, A033950, A057265, A049439.
%K A036436 easy,nonn
%O A036436 1,2
%A A036436 Simon Colton (simonco(AT)cs.york.ac.uk)
%E A036436 Links corrected and edited by _Daniel Forgues_, Jun 30 2010