A389498 - OEIS

A389498

Powerful numbers whose number of powerful divisors is a square.

1, 16, 36, 81, 100, 196, 216, 225, 441, 484, 512, 625, 676, 1000, 1089, 1156, 1225, 1296, 1444, 1521, 2116, 2304, 2401, 2601, 2744, 3025, 3249, 3364, 3375, 3600, 3844, 4225, 4761, 5476, 5929, 6400, 6724, 7056, 7225, 7396, 7569, 7776, 8100, 8281, 8649, 8836, 9025, 9261, 10000

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

COMMENTS

Powerful numbers k for which the product of their prime factorization exponents, A005361(k), is a square.

Primitive terms in A389497: If k is a term, then k*m is a term in A389497 for any squarefree number m that is coprime to k.

Any term of A389497 is of the form k*m where k is a term in this sequence and m is a squarefree number coprime to k. Therefore, A389497 can be generated from this sequence by multiplying with coprime squarefree numbers, and the asymptotic density of A389497 can be evaluated from the terms in this sequence (see the Comments section of A389497).

LINKS

Amiram Eldar, Table of n, a(n) for n = 1..10000

MATHEMATICA

pows[max_] := Union[Flatten[Table[i^2*j^3, {j, 1, Surd[max, 3]}, {i, 1, Sqrt[max/j^3]}]]];

q[k_] := IntegerQ[Sqrt[Times @@ FactorInteger[k][[;; , 2]]]];

seq[max_] := Select[pows[max], q]; seq[10000]

PROG

(PARI) is1(k) = issquare(vecprod(factor(k)[, 2]));

list(lim) = {my(s = List(), m); for(j = 1, sqrtnint(lim, 3), for(i = 1, sqrtint(lim\j^3), m = i^2*j^3; if(is1(m), listput(s, m)))); Set(s); }

CROSSREFS

Intersection of A001694 and A389497.

Subsequence of A389500.