A391086 Primitive exponential abundant numbers that are not primitives exponential unitary abundant.

3600, 7056, 8100, 435600, 529200, 608400, 853776, 882000, 1040400, 1192464, 1234800, 1299600, 1587600, 1904400, 1920996, 1984500, 2039184, 2178000, 2340900, 2371600, 2547216, 2683044, 2778300, 2924100, 2944656, 3027600, 3042000, 3312400, 3459600, 3732624, 3920400

Offset

1,1

Comments

The powerful terms in A391085.

There are no terms that are powerful exponentially squarefree numbers (A384418) because there are no exponentially squarefree numbers (A209061) in A391085.

Any term of A391085 is of the form k*m where k is a term of this sequence and m is a squarefree number coprime to k. Therefore, A391085 can be generated from this sequence by multiplying terms with coprime squarefree numbers, and the asymptotic density of A391085 can be evaluated from the terms of this sequence (see the Comments section of A391085).

The least odd term is a(3629) = 38101088025 = (3 * 5 * 7 * 11 * 13^2)^2.

Links

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

Mathematica

f1[p_, e_] := DivisorSum[e, p^# &];
f2[p_, e_] := DivisorSum[e, p^# &, GCD[#, e/#] == 1 &];
q[k_] := Module[{f = FactorInteger[k]}, Times @@ f1 @@@ f > 2*k && Times @@ f2 @@@ f <= 2*k];
seq[lim_] := Select[Union[Flatten[Table[i^2 * j^3, {j, 1, Surd[lim, 3]}, {i, 1, Sqrt[lim/j^3]}]]], q]; seq[4*10^6]

Prog

(PARI) fun1(p, e) = sumdiv(e, d, p^d);
fun2(p, e) = sumdiv(e, d, if(gcd(d, e/d) == 1, p^d));
is1(k) = {my(f = factor(k)); prod(i = 1, #f~, fun1(f[i, 1], f[i, 2])) > 2*k && prod(i = 1, #f~, fun2(f[i, 1], f[i, 2])) <= 2*k; }
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 A391085.

Cf. A005117, A209061, A384418.

Sequence in context: A391090 A391091 A391085 * A096472 A306492 A364990

Adjacent sequences: A391083 A391084 A391085 * A391087 A391088 A391089

Keyword

nonn,easy

Author

Amiram Eldar, Nov 28 2025

Status

approved