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

1,2

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

Links

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

Formula

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

Example

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

Mathematica

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

Prog

(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

Crossrefs

Cf. A000290, A048691.

Subsequences: A030229, A238748.

Sequence in context: A129146 A315158 A315159 * A369256 A068993 A138592

Adjacent sequences: A063771 A063772 A063773 * A063775 A063776 A063777

Keyword

easy,nonn

Author

Jason Earls, Aug 15 2001

Status

approved