For N of the forms 16n+14, 36n+30, 72n+42, 196n+70 and 252n+114, the sum over k < sqrt(N) of sigma_0(N - k^2) is congruent to 0 mod 4.
Hooley , On the representation of a number as the sum of a square and a product , Mathematische Zeitschrift, 69 (1958), pp
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Congruences via Partitions with Exactly Two Part Sizes
For N of the forms 16n+14, 36n+30, 72n+42, 196n+70 and 252n+114, the sum over k < sqrt(N) of sigma_0(N - k^2) is congruent to 0 mod 4.