OFFSET
1,1
COMMENTS
First differs from its subsequence A389298 at n = 17: a(17) = 17640 = 2^3 * 3^2 * 5 * 7^2 is not a term of A389298.
Exponential deficient numbers are numbers k with esigma(k)/k < 2, where esigma = A051377. These are numbers that are neither e-perfect (A054979) nor exponential abundant (A129575).
An exponential 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 (A389298) 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(A389298(n)) = 0.0013366..., where f(n) = (6/(Pi^2*n)) * Product_{prime p|n} (p/(p+1)).
The least odd term is a(904037) = 676350675.
LINKS
Amiram Eldar, Table of n, a(n) for n = 1..10000
MATHEMATICA
f[p_, e_] := DivisorSum[e, p^# &]; esigma[1] = 1; esigma[n_] := Times @@ f @@@ FactorInteger[n];
expAbQ[n_] := esigma[n] > 2*n; expDefQ[n_] := esigma[n] < 2*n; q[n_] := expDefQ[n] && AnyTrue[Divisors[n], expAbQ]; Select[Range[36000], q]
PROG
(PARI) isok(k) = {my(f = factor(k)); prod(i = 1, #f~, sumdiv(f[i, 2], d, f[i, 1]^d)) < 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 29 2025
STATUS
approved