OFFSET
1,1
COMMENTS
Intersection of A390539 (noncubefull Achilles numbers) and A392134 (Achilles numbers that are neither 4-free nor 4-full).
Intersection of A362147 (noncubefull numbers), A052486 (Achilles numbers), and A391115 (numbers that are neither 4-free nor 4-full).
Let omega = A001221 be the number of distinct prime factors p | k. Let p^m | k but p^(m+1) not divide k and define Q to be the multiset of m for all p | k. Define c(Q) to be the number of m such that m >= 4. This sequence is {k : gcd(Q) = 1, min(Q) = 2, 0 < c(Q) < omega(k)}.
LINKS
Michael De Vlieger, Table of n, a(n) for n = 1..10000
EXAMPLE
Table n, a(n), n = 1..12:
n a(n)
----------------------------
1 288 = 2^5 * 3^2
2 800 = 2^5 * 5^2
3 972 = 2^2 * 3^5
4 1152 = 2^7 * 3^2
5 1568 = 2^5 * 7^2
6 3200 = 2^7 * 5^2
7 3872 = 2^5 * 11^2
8 4608 = 2^9 * 3^2
9 5408 = 2^5 * 13^2
10 6075 = 3^5 * 5^2
11 6272 = 2^7 * 7^2
12 7200 = 2^5 * 3^2 * 5^2
MATHEMATICA
nn = 40000; Union@ Flatten@ Table[If[And[Min[#2] == 2, 0 < Count[#2, _?(# > 3 &)] < Length[#2], GCD @@ #2 == 1], #1, Nothing] & @@ {#, FactorInteger[#][[;; , -1]]} &[a^2*b^3], {b, Surd[nn, 3]}, {a, Sqrt[nn/b^3]}]
PROG
(PARI) is_A392907(n) = if(n<=1, 0, (e->(1==gcd(e) && vecmin(e)==2 && vecmax(e)>3))(factor(n)[, 2])); \\ Antti Karttunen, Jan 26 2026
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Michael De Vlieger, Jan 26 2026
STATUS
approved