OFFSET
1,1
COMMENTS
Subsequence of A389330 and first differs from it at n = 21: A389330(21) = 17640 = 2^3 * 3^2 * 5 * 7^2 is not a term of this sequence.
Any term of A389330 is of the form k*m where k is a term of this sequence and m is a squarefree number coprime to k. Therefore, A389330 can be generated from this sequence by multiplying with coprime squarefree numbers, and the asymptotic density of A389330 can be evaluated from the terms of this sequence (see the Comments section of A389330).
The least odd term is a(13652) = 676350675.
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]}]]];
f[p_, e_] := DivisorSum[e, p^# &, GCD[#, e/#] == 1 &]; eusigma[1] = 1; eusigma[n_] := Times @@ f @@@ FactorInteger[n];
expUAbQ[n_] := eusigma[n] > 2*n; expUDefQ[n_] := eusigma[n] < 2*n; q[n_] := expUDefQ[n] && AnyTrue[Divisors[n], expUAbQ];
seq[max_] := Select[pows[max], q]; seq[35000]
PROG
(PARI) fun(p, e) = sumdiv(e, d, if(gcd(d, e/d) == 1, p^d));
is1(k) = {my(f = factor(k)); prod(i = 1, #f~, fun(f[i, 1], f[i, 2])) < 2*k && prod(i = 1, #f~, if(f[i, 2] > 1, 1+1/f[i, 1], 1)) > 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
KEYWORD
nonn
AUTHOR
Amiram Eldar, Sep 30 2025
STATUS
approved