OFFSET
1,1
COMMENTS
Numbers m with coreful divisors d, m/d such that neither d | m/d nor m/d | d, i.e., numbers m such that there exists a divisor pair (d, m/d) such that rad(d) = rad(m/d) but gcd(d, m/d) > 1 is neither d nor m/d, where rad = A007947. Divisors in each pair must be dissimilar and each in A126706.
Proper subset of A320966.
LINKS
Michael De Vlieger, Table of n, a(n) for n = 1..10000
Michael De Vlieger, Notes on this sequence
FORMULA
Sum_{n>=1} 1/a(n) = zeta(2)*zeta(3)/zeta(6) - (15/Pi^2) * (1 + Sum_{prime} 1/((p-1)*(p^2+1))) = 0.021194288968234037106579437374641326044... . - Amiram Eldar, Nov 08 2024
EXAMPLE
216 is in the sequence since rad(12) | rad(18), but 12 does not divide 18 and 18 does not divide 12.
432 is a term since rad(18) | rad(24), but 18 does not divide 24 and 24 does not divide 18.
Table of coreful divisors d, a(n)/d such that neither d | a(n)/d nor a(n)/d | d for select a(n)
n | a(n) divisor pairs d X a(n)/d
---------------------------------------------------------------------------
1 | 216: 12 X 18;
2 | 432: 18 X 24;
3 | 648: 12 X 54;
4 | 864: 24 X 36, 18 X 48;
5 | 1000: 20 X 50;
6 | 1296: 24 X 54;
7 | 1728: 18 X 96, 36 X 48;
8 | 1944: 12 X 162, 36 X 54;
9 | 2000: 40 X 50;
10 | 2592: 24 X 108, 48 X 54;
11 | 2744: 28 X 98;
12 | 3375: 45 X 75;
13 | 3456: 18 X 192, 36 X 96, 48 X 72;
22 | 7776: 24 X 324, 48 X 162, 54 X 144, 72 X 108;
58 | 31104: 48 X 648, 54 X 576, 96 X 324, 108 X 288, 144 X 216, 162 X 192
MATHEMATICA
Union@ Select[
Flatten@ Table[a^2*b^3, {b, Surd[#, 3]}, {a, Sqrt[#/b^3]}] &[20000],
Length@ Select[FactorInteger[#][[All, -1]], # > 2 &] >= 2 &]
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Michael De Vlieger, Oct 16 2024
STATUS
approved