A063774 Numbers k such that the number of divisors of k^2 is a square.

1, 6, 10, 14, 15, 16, 21, 22, 26, 33, 34, 35, 36, 38, 39, 46, 51, 55, 57, 58, 62, 65, 69, 74, 77, 81, 82, 85, 86, 87, 91, 93, 94, 95, 100, 106, 111, 115, 118, 119, 122, 123, 129, 133, 134, 141, 142, 143, 145, 146, 155, 158, 159, 161, 166, 177, 178, 183, 185, 187, 194

Offset: 1

Author

Jason Earls, Aug 15 2001

Keywords

• easy
• nonn

Comments

The numbers of terms not exceeding 10^k, for k = 1, 2, ..., are 3, 35, 326, 3275, 33090, 332435, 3327555, 33283964, 332868092, 3328794682, ... . Apparently, the asymptotic density of this sequence exists and equals 0.3328... . - Amiram Eldar, Nov 28 2023

Examples

			n=2: a(2) = 6 because the number of divisors of 6^2 is 9, a square.
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Harry J. Smith)

Crossrefs

Cf. A000290, A048691.

Subsequences: A030229, A238748.

Programs

Mathematica

Select[Range[200],IntegerQ[Sqrt[DivisorSigma[0,#^2]]]&] (* Harvey P. Dale, Jun 06 2012 *)

PARI

j=[]; for(n=1,500,a=numdiv(n^2); if(issquare(a),j=concat(j,n))); j

PARI

n=0; for (m=1, 10^9, if(issquare(numdiv(m^2)), write("b063774.txt", n++, " ", m); if (n==1000, break))) \\ Harry J. Smith, Aug 30 2009

PARI

is(n)=my(f=factor(n)[,2]); issquare(prod(i=1,#f,2*f[i]+1)) \\ Charles R Greathouse IV, Sep 18 2015

Formula

{n: A048691(n) in A000290}. - R. J. Mathar, Aug 09 2012