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A230902
Positive numbers such that half of the set of divisors are of the form x^2 + x*y + y^2 (A003136) and half not (A034020).
0
2, 5, 6, 8, 11, 14, 15, 17, 18, 23, 24, 26, 29, 32, 33, 35, 38, 41, 42, 45, 47, 51, 53, 54, 56, 59, 62, 65, 69, 71, 72, 74, 77, 78, 83, 86, 87, 89, 95, 96, 98, 99, 101, 104, 105, 107, 113, 114, 119, 122, 123, 125, 126, 128, 131, 134, 135, 137, 141, 143, 146, 149, 152, 153, 155, 158, 159, 161, 162
OFFSET
1,1
EXAMPLE
Triangle read by rows in which row n lists the divisors of n begins:
1(0^2+0*1+1^2);
1(0^2+0*1+1^2), 2;
1(0^2+0*1+1^2), 3(1^1+1*1+1^2);
1(0^2+0*1+1^2), 2, 4(0^2+0*2+2^2);
1(0^2+0*1+1^2), 5;
1(0^2+0*1+1^2), 2, 3(1^1+1*1+1^2), 6;
1(0^2+0*1+1^2), 7(1^1+1*2+2^2);
1(0^2+0*1+1^2), 2, 4(0^2+0*2+2^2), 8;
1(0^2+0*1+1^2), 3(1^1+1*1+1^2), 9;
1(0^2+0*1+1^2), 2, 5, 10;
1(0^2+0*1+1^2), 11;
1(0^2+0*1+1^2), 2, 3(1^1+1*1+1^2), 4(0^2+0*2+2^2), 6, 12(2^2+2*2+2^2);
1(0^2+0*1+1^2), 13(1^2+1*3+3^2);
1(0^2+0*1+1^2), 2, 7(1^1+1*2+2^2), 14;
1(0^2+0*1+1^1), 3(1^11+1*1+1^2), 5, 15,
i.e. a(1)=2, a(2)=5, a(3)=6, a(4)=8, a(5)=11, a(6)=14, a(7)=15.
MAPLE
isA003136 := proc(n)
local x, y ;
for x from 0 do
if x^2 > n then
return false;
end if;
for y from 0 do
if x^2+x*y+y^2 = n then
return true;
elif x^2+x*y+y^2 > n then
break;
end if;
end do:
end do:
end proc:
isA230902 := proc(n)
local a36, a20, d ;
a36 := 0 ;
a20 := 0 ;
for d in numtheory[divisors](n) do
if isA003136(d) then
a36 := a36+1 ;
else
a20 := a20+1 ;
end if;
end do:
simplify( a36=a20) ;
end proc:
for n from 0 to 200 do
if isA230902(n) then
printf("%d, ", n);
end if;
end do: # R. J. Mathar, Nov 08 2013
MATHEMATICA
A003136Q[n_] := Resolve[Exists[{x, y}, Reduce[n == x^2 + x*y + y^2, {x, y}, Integers]]];
okQ[n_] := With[{dd = Divisors[n]}, 2 Count[dd, _?A003136Q] == Length[dd]];
Select[Range[200], okQ] (* Jean-François Alcover, Jun 07 2024 *)
CROSSREFS
Cf. A027750, A230851. Subsequence of A000037.
Sequence in context: A214642 A026179 A300063 * A243680 A280381 A333965
KEYWORD
nonn
AUTHOR
EXTENSIONS
Corrected by R. J. Mathar, Nov 08 2013
STATUS
approved