OFFSET
1,1
COMMENTS
First differs from its subsequence A389331 at n = 21: a(21) = 17640 = 2^3 * 3^2 * 5 * 7^2 is not a term of A389331.
Exponential unitary deficient numbers are numbers k with esigma(k)/k < 2, where esigma = A051377. These are numbers that are neither e-unitary perfect (A054979 \ A322858) nor exponential unitary abundant (A383693).
An exponential unitary deficient number k is a term if and only if Product_{primes p, p^2 | k} (1+1/p) > 2.
All the terms are noncubefree (A046099).
If k is a term and m is a squarefree number coprime to k, then k*m is also a term. The primitive terms of this sequence (A389331) 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(A389331(n)) = 0.0014992..., where f(n) = (6/(Pi^2*n)) * Product_{prime p|n}(p/(p+1)).
The least odd term is a(1014006) = 676350675.
LINKS
Amiram Eldar, Table of n, a(n) for n = 1..10000
MATHEMATICA
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]; Select[Range[30000], q]
PROG
(PARI) fun(p, e) = sumdiv(e, d, if(gcd(d, e/d) == 1, p^d));
isok(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; }
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Amiram Eldar, Sep 30 2025
STATUS
approved