On the Diophantine equations of the form ${\lambda }_1{U}_{n_1}+{\lambda }_2{U}_{n_2}+\dots +{\lambda }_k{U}_{n_k}=w{p}_1^{z_1}{p}_2^{z_2}\cdots p_s^{z_s}$

In this paper, we consider the Diophantine equation ${\lambda }_1{U}_{n_1}+\dots +{\lambda }_k{U}_{n_k}=w{p}_1^{z_1}\cdots p_s^{z_s},$ where $\{U_n{\}}_{n\ge 0}$ is a fixed non-degenerate linear recurrence sequence of order greater than or equal to 2; $w$ is a fixed non-zero integer; $p_1,\dots ,p_s$ are fixed, distinct prime numbers; ${\lambda }_1,\dots ,{\lambda }_k$ are strictly positive integers; and $n_1,\dots ,n_k,z_1,\dots ,z_s$ are non-negative integer unknowns. We prove the existence of an effectively computable upper-bound on the solutions $(n_1,\dots ,n_k,z_1,\dots ,z_s)$. In our proof, we use lower bounds for linear forms in logarithms, extending the work of Pink and Ziegler (2016), Mazumdar and Rout (2019), Meher and Rout (2017), and Ziegler (2019).