Abstract
Given a set of positive integers \(A=(a_1, a_2, \ldots , a_n)\) whose greatest common divisor is 1, the Frobenius number g(A) is the largest integer not representable as a linear combination of the \(a_i\)’s with nonnegative integer coefficients. We find the stable property introduced for the square sequence \(A=(a,a+1,a+2^2,\dots , a+k^2)\) naturally extends for \(A(a)=(a,ha+dB)=(a,ha+d,ha+b_2d, \ldots ,ha+b_kd)\). This gives a parallel characterization of g(A(a)) as a "congruence class function" modulo \(b_k\) when a is large enough. For orderly sequence \(B=(1,b_2,\dots ,b_k)\), we find good bound for a. In particular we calculate \(g(a,ha+dB)\) for \(B=(1,2,b,b+1)\), \(B=(1,2,b,b+1,2b)\), \(B=(1,b,2b-1)\), and \(B=(1,2, \ldots ,k,K)\).
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Liu, F., Xin, G., Ye, S. et al. The Frobenius formula for \(A=(a,ha+d,ha+b_2d, \ldots ,ha+b_kd)\). Ramanujan J 64, 489–504 (2024). https://doi.org/10.1007/s11139-024-00837-2
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DOI: https://doi.org/10.1007/s11139-024-00837-2