OFFSET
1,1
COMMENTS
From Amiram Eldar, Dec 01 2025: (Start)
If k is a term and m is a squarefree number coprime to k, then k*m is also a term.
There are two types of terms in this sequence:
1) Terms of the form k*m where k is a powerful (A001694) term in this sequence (A391140) and m is a squarefree number coprime to k.
2) Terms of the form k*m where k is in A391141, i.e., a powerful number that is coreful abundant with an odd sum of coreful divisors (which is a powerful number in the intersection of A308053 and A376218), and m is an even squarefree number coprime to k.
The least term of type 2 is a(17302) = 2315250.
The asymptotic density of this sequence is Sum_{n>=1} f(A391140(n)) + Sum_{n>=1} f(A391141(n))/3 = 0.007468..., where f(n) = (6/(Pi^2*n)) * Product_{prime p|n} (p/(p+1)), and the two sums correspond to the two types above. The asymptotic density of the terms of the second type is 1.202286997...*10^(-7). (End)
LINKS
Amiram Eldar, Table of n, a(n) for n = 1..10000
EXAMPLE
108 is a term since its set of coreful divisors, {6, 12, 18, 36, 54, 108}, has an even sum, 234 > 2*108, and it cannot be partitioned into two disjoint sets of equal sum.
MATHEMATICA
q[n_] := Module[{r = Times @@ FactorInteger[n][[;; , 1]], d, sum, x}, d = r*Divisors[n/r]; (sum = Plus @@ d) >= 2*n && EvenQ[sum] && CoefficientList[Product[1 + x^i, {i, d}], x][[1 + sum/2]] == 0]; Select[Range[10^4], q]
KEYWORD
nonn
AUTHOR
Amiram Eldar, Dec 25 2020
STATUS
approved