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Let $ S $ be a given finite set of positive and relatively prime integers. Denote $ L(S) $ to be the set of integers obtained by taking all nonnegative integer linear combinations of integers in $ S $. It is well known that there are finitely many positive integers that are not in $ L(S) $. Let $ g(S) $ and $ n(S) $ represent the greatest integer that does not belong to $ L(S) $ and the number of nonnegative integers that do not belong to $ L(S) $, respectively. The Frobenius problem is to determine $ g(S) $ and $ n(S) $. In 2016, Tripathi obtained results on $ g(S) $ and $ n(S) $ when $ S = \{a, ha+d, ha+db, ha+db^2, \ldots, ha+db^k\} $. In this paper, for $ S_c: = \{a, ha+d, ha+c+db, ha+2c+db^2, \ldots, ha+kc+db^k\} $ with $ h, c $ being nonnegative integers, $ a, b, d $ being positive integers and $ \gcd(a, d) = 1 $, we focused the investigation on formulas for $ g(S_c) $ and $ n(S_c) $. Actually, we gave formulas for $ g(S_c) $ and $ n(S_c) $ for all sufficiently large values of $ d $ when $ c $ is any multiple of $ d $ or certain multiples of $ a $. This generalized the results of Tripathi in 2016.

All \(\mathcal{VB}_{m,n}\)-gauge-natural operators \(C\) sending linear \(3\)-forms \(H \in \Gamma^{l}_E(\bigwedge^3T^*E)\) on a smooth (\(\mathcal{C}^\infty\)) vector bundle \(E\) into \(\mathbf{R}\)-bilinear operators \[C_H:\Gamma^l_E(TE \oplus T^*E)\times \Gamma^l_E(TE \oplus T^*E)\to \Gamma^l_E(TE \oplus T^*E)\] transforming pairs of linear sections of \(TE \oplus T^*E \to E\) into linear sections of \( TE \oplus T^*E \to E\) are completely described. The complete descriptions is given of all generalized twisted Dorfman-Courant brackets \(C\) (i.e. \(C\) as above such that \(C_0\) is the Dorfman-Courant bracket) satisfying the Jacobi identity for closed linear \(3\)-forms \(H\). An interesting natural characterization of the (usual) twisted Dorfman-Courant bracket is presented.

There are completely described all \(\mathcal{VB}_{m,n}\)-gauge-natural operators \(C\) which, like to the Dorfman-Courant bracket, send closed linear \(3\)-forms \(H\in\Gamma^{l-\rm{clos}}_E(\bigwedge^3T^*E)\) on a smooth (\(\mathcal{C}^{\infty}\)) vector bundle \(E\) into \(\mathbf{R}\)-bilinear operators \[C_H:\Gamma^l_E(TE\oplus T^*E)\times \Gamma^l_E(TE\oplus T^*E)\to \Gamma^l_E(TE\oplus T^*E)\] transforming pairs of linear sections of \(TE\oplus T^*E\to E\) into linear sections of \(TE\oplus T^*E\to E\). Then all such \(C\) which also, like to the twisted Dorfman-Courant bracket, satisfy both some "restricted" condition and the Jacobi identity in Leibniz form are extracted.