Journal of Integer Sequences, Vol. 16 (2013), Article 13.7.4

## The Frobenius Problem for Modified Arithmetic Progressions

### Amitabha Tripathi Department of Mathematics Indian Institute of Technology Hauz Khas, New Delhi – 110016 India

Abstract:

For a set of positive and relative prime integers A, let Γ(A) denote the set of integers obtained by taking all nonnegative integer linear combinations of integers in A. Then there are finitely many positive integers that do not belong to Γ(A). For the modified arithmetic progression A = {a, ha + d, ha + 2d, ... , ha + kd}, gcd(a, d) = 1, we determine the largest integer g(A) that does not belong to Γ(A), and the number of integers n(A) that do not belong to Γ(A). We also determine the set of integers S*(A) that do not belong to Γ(A) which, when added to any positive integer in Γ(A), result in an integer in Γ(A). Our results generalize the corresponding results for arithmetic progressions.

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Received May 15 2013; revised version received July 31 2013. Published in Journal of Integer Sequences, August 1 2013.