Geodesic
The Hardy--Littlewood problem for numbers with a~fixed number of prime divisors
Izvestiya. Mathematics , Tome 59 (1995) no. 6, pp. 1283-1309.

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In this paper we investigate the number of representations of a natural number $N$ as the sum of a number with $k$ prime divisors and two squares, where $k$ may depend on $N$. We determine the asymptotic behaviour when $2\leqslant k\leqslant(2-\varepsilon)\ln\ln N$ and $(2+\varepsilon)\ln\ln N\leqslant k\leqslant b\ln\ln N$. 
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N. M. Timofeev. The Hardy--Littlewood problem for numbers with a~fixed number of prime divisors. Izvestiya. Mathematics , Tome 59 (1995) no. 6, pp. 1283-1309. http://geodesic.mathdoc.fr/item/IM2_1995_59_6_a8/

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