A391085 Exponential abundant numbers that are not exponential unitary abundant.

3600, 7056, 8100, 25200, 35280, 39600, 46800, 56700, 61200, 68400, 77616, 82800, 89100, 91728, 104400, 105300, 111600, 119952, 133200, 134064, 137700, 147600, 153900, 154800, 162288, 169200, 186300, 190800, 204624, 212400, 218736, 219600, 234900, 241200, 251100

Offset

1,1

Comments

Numbers k such that A322857(k) <= 2*k < A051377(k).

If k is an exponentially squarefree number (A209061), then the sets of exponential divisors and exponential unitary divisors of k coincide and A051377(k) = A322857(k). Therefore, all the terms are not exponentially squarefree numbers (A130897).

All the exponential unitary abundant numbers are also exponential abundant numbers. Therefore, this sequence is the complement of A383693 within A129575.

If k is a term and m is a squarefree number coprime to k, then k*m is also a term. The primitive terms in this sequence (A391086) are the powerful (A001694) terms. All the terms are of the form k*m where k is primitive and m is a squarefree number coprime to k.

The asymptotic density of this sequence is Sum_{n>=1} f(A391086(n)) = 0.0001464..., where f(n) = (6/(Pi^2*n))*Product_{prime p|n}(p/(p+1)).

The least odd term is a(5579933) = 38101088025 = (3 * 5 * 7 * 11 * 13^2)^2.

Links

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

Formula

Equals A129575 \ A383693.

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];
Select[Range[3*10^5], q]

Prog

(PARI) fun1(p, e) = sumdiv(e, d, p^d);
fun2(p, e) = sumdiv(e, d, if(gcd(d, e/d) == 1, p^d));
isok(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; }

Crossrefs

Subsequence of A129575 and A130897.

A391086 is a subsequence.

Cf. A001694, A005117, A129575, A130897, A209061, A322857, A383693.

Sequence in context: A179746 A391090 A391091 * A391086 A096472 A306492

Adjacent sequences: A391082 A391083 A391084 * A391086 A391087 A391088

Keyword

nonn,easy

Author

Amiram Eldar, Nov 28 2025

Status

approved