OFFSET
1,6
COMMENTS
Conjecture: a(n) > 0 for all n > 5
This has been verified for n <= 150000. - Robert G. Wilson v, Jan 28 2026
LINKS
Robert G. Wilson v, Table of n, a(n) for n = 1..10000 (first 5000 terms from Zhi-Wei Sun)
EXAMPLE
a(3) = 1 since 3 = 1^2 + 1 + 1 with lcm(1^2,1,1) = 1^2.
a(8) = 1 since 8 = 2^2 + 2 + 2 with lcm(2^2,2,2) = 2^2.
a(10) = 1 since 10 = 2^2 + 4 + 2 with lcm(2^2,4,2) = 2^2.
a(12) = 1 since 12 = 2^2 + 4 + 4 with lcm(2^2,4,4) = 2^2.
a(20) = 1 since 20 = 4^2 + 2 + 2 with lcm(4^2,2,2) = 4^2.
a(32) = 1 since 32 = 4^2 + 8 + 8 with lcm(4^2,8,8) = 4^2.
a(80) = 1 since 80 = 8^2 + 8 + 8 with lcm(8^2,8,8) = 8^2.
MATHEMATICA
SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]];
tab={}; Do[r=0; Do[q=LCM[k^2, m, n-k^2-m]; If[SQ[q], r=r+1], {k, 1, Sqrt[n]}, {m, 1, (n-k^2)/2}]; tab=Append[tab, r], {n, 1, 100}]; Print[tab]
a = Compile[{{n, _Integer}}, Block[{c = 0}, Do[ c += Boole[ Mod[ Sqrt[ LCM[ s^2, x, n -s^2 -x]], 1] == 0], {s, Sqrt[n]}, {x, (n -s^2)/2}]; c]]; Array[a, 86] (* Robert G. Wilson v, Jan 28 2026 *)
CROSSREFS
KEYWORD
nonn
AUTHOR
Zhi-Wei Sun, Jan 17 2026
STATUS
approved
