OFFSET
1,1
COMMENTS
Also, the squares in A376936.
Proper subset of A378767, in turn a proper subset of A286708, the intersection of A001694 and A024619.
Numbers that have 3 kinds of coreful divisor pairs (d, k/d), d | k, i.e., rad(d) = rad(k/d) = rad(k) where rad = A007947. These kinds are described as follows:
Type A: d = k/d, which pertain to square k (in A000290).
Type B: d | k/d, d < k/d, which pertain to k in A320966, powerful numbers divisible by a cube.
Type C: neither d | k/d nor k/d | d, which pertain to k in A376936.
Since divisors d, k/d may either divide or not divide the other, there are no other cases.
In addition the following kinds of divisor pairs are also seen:
Type D: (d, k/d) such that d | k/d but there exists a factor Q | k/d that does not divide d. Then omega(d) < omega(k/d) = omega(k).
Type E: Nontrivial unitary divisor pairs (d, k/d) such that gcd(d, k/d) = 1, d > 1, k/d > 1. Let prime power factor p^m | k be such that m is maximized. Then set d = p^m and it is clear that for any k in A024619, there exists at least 1 nontrivial unitary divisor pair.
LINKS
Michael De Vlieger, Table of n, a(n) for n = 1..10000
Michael De Vlieger, Listing of select divisor pairs of a(n), n = 1..12, showing divisor pairs of type A in light gray, type B in orange and purple, and type C in black.
FORMULA
a(n) = A036785(n)^2.
EXAMPLE
Let b = A036785.
Table of the first 12 terms of this sequence, showing examples of types A, B, and C of coreful pairs of divisors.
n a(n) Factors of a(n) b(n) Type B Type C
-------------------------------------------------------------
1 1296 2^4 * 3^4 36 6 * 216 24 * 54
2 5184 2^6 * 3^4 72 6 * 864 48 * 108
3 10000 2^4 * 5^4 100 10 * 1000 40 * 250
4 11664 2^4 * 3^6 108 6 * 1944 24 * 486
5 20736 2^8 * 3^4 144 6 * 3456 54 * 384
6 32400 2^4 * 3^4 * 5^2 180 30 * 1080 120 * 270
7 38416 2^4 * 7^4 196 14 * 2744 56 * 686
8 40000 2^6 * 5^4 200 10 * 4000 80 * 500
9 46656 2^6 * 3^6 216 6 * 7776 48 * 972
10 50625 3^4 * 5^4 225 15 * 3375 135 * 375
11 63504 2^4 * 3^4 * 7^2 252 42 * 1512 168 * 378
12 82944 2^10 * 3^4 288 6 * 13824 54 * 1536
MATHEMATICA
s = Union@ Select[Flatten@ Table[a^2*b^3, {b, Surd[#, 3]}, {a, Sqrt[#/b^3]}], IntegerQ@ Sqrt[#] &] &[500000];
Union@ Select[s, Length@ Select[FactorInteger[#][[All, -1]], # > 2 &] >= 2 &]
CROSSREFS
KEYWORD
nonn,easy,new
AUTHOR
Michael De Vlieger, Dec 12 2024
STATUS
approved