Geodesic
Additive problems with numbers having a given number of prime dividers from progressions
Fundamentalʹnaâ i prikladnaâ matematika, Tome 3 (1997) no. 1, pp. 163-170

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We have found the number of the representations of a number $N$ as $$ n=mr\quadand\quad n+m^2+r^2, $$ where $m,r$ — natural numbers and $n$ are the numbers having $k$ prime dividers such that $p_i\equiv l_i\, (\bmod\ d_0)$, $p_i\geq t> \ln^{B+1}N$, $(l_i,d_0)=1$, $i=1,2,\ldots,k$, $(N-l_1\ldots l_k,d_0)=1$. The paper also contains the results about distribution of such numbers $n$ in arithmetic progressions with large modulus. 
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     author = {A. A. Zhukova},
     title = {Additive problems with numbers having a given number of prime dividers from progressions},
     journal = {Fundamentalʹna\^a i prikladna\^a matematika},
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     year = {1997},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/FPM_1997_3_1_a13/}
}
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A. A. Zhukova. Additive problems with numbers having a given number of prime dividers from progressions. Fundamentalʹnaâ i prikladnaâ matematika, Tome 3 (1997) no. 1, pp. 163-170. http://geodesic.mathdoc.fr/item/FPM_1997_3_1_a13/