Is this number Loeschian?
Jelly, 11 9 bytes
ÆF‘%3,2ḄȦ

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Background
In Elementary results on the binary quadratic form a² + ab + b², the author proves the following theorem about Löschian numbers.

Theorem 16. The necessary and sufficient condition of any non-negative integer to be in the form a² + ab + b² is that, in its prime factorization, all primes other than 3 that are not in the form (6k + 1) have even exponents.

As noted on the relevant OEIS page, since all integers are are congruent to 0, 1 or 2 modulo 3, the number 3 is the only prime that is congruent to 0, and all numbers of the form (6k + 1) are congruent to 1, the theorem can be stated alternatively as follows.
A non-negative integer n is a Löschian number if and only if all prime factors of n that are congruent to 2 modulo 3 have even exponents.
How it works
ÆF‘%3,2ḄȦ  Main link. Argument: n (integer)

ÆF         Yield the prime factorization of n, as prime-exponent pairs.
  ‘        Increment all primes and exponents, turning primes of the form 3k - 2
           into multiples of 3 and odd exponents into multiples of 2.
   %3,2    Reduce all incremented primes/exponents modulo 3/2.
           n is Löschian if and only if this does not result in a [0, 0] pair.
           Due to Jelly's form of vectorization, this yields [3, 2] if n = 1.
       Ḅ   Unbinary; convert each pair from base 2 to integer.
           Note that [x, y] = [0, 0] if and only if 2x + y = 0.
        Ȧ  All; return 1 if the result contains no zeroes, 0 otherwise.