Geodesic
The postage stamp problem and arithmetic in base $r$
Czechoslovak Mathematical Journal, Tome 58 (2008) no. 4, pp. 1097-1100 Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

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Let $h,k$ be fixed positive integers, and let $A$ be any set of positive integers. Let $hA:=\{a_1+a_2+\cdots +a_r\colon a_i \in A, r \le h\}$ denote the set of all integers representable as a sum of no more than $h$ elements of $A$, and let $n(h,A)$ denote the largest integer $n$ such that $\{1,2,\ldots ,n\} \subseteq hA$. Let $n(h,k):=\max _A\colon n(h,A)$, where the maximum is taken over all sets $A$ with $k$ elements. We determine $n(h,A)$ when the elements of $A$ are in geometric progression. In particular, this results in the evaluation of $n(h,2)$ and yields surprisingly sharp lower bounds for $n(h,k)$, particularly for $k=3$.
Let $h,k$ be fixed positive integers, and let $A$ be any set of positive integers. Let $hA:=\{a_1+a_2+\cdots +a_r\colon a_i \in A, r \le h\}$ denote the set of all integers representable as a sum of no more than $h$ elements of $A$, and let $n(h,A)$ denote the largest integer $n$ such that $\{1,2,\ldots ,n\} \subseteq hA$. Let $n(h,k):=\max _A\colon n(h,A)$, where the maximum is taken over all sets $A$ with $k$ elements. We determine $n(h,A)$ when the elements of $A$ are in geometric progression. In particular, this results in the evaluation of $n(h,2)$ and yields surprisingly sharp lower bounds for $n(h,k)$, particularly for $k=3$.
Classification : 11B13, 11D04
Keywords: $h$-basis; extremal $h$-basis; geometric progression 
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     title = {The postage stamp problem and arithmetic in base $r$},
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}
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Tripathi, Amitabha. The postage stamp problem and arithmetic in base $r$. Czechoslovak Mathematical Journal, Tome 58 (2008) no. 4, pp. 1097-1100. http://geodesic.mathdoc.fr/item/CMJ_2008_58_4_a15/

[1] Alter, R., Barnett, J. A.: A postage stamp problem. Amer. Math. Monthly 87 206-210 (1980). | DOI | MR | Zbl

[2] Hofmeister, G.: Asymptotische Abschätzungen für dreielementige Extremalbasen in natürlichen Zahlen. J. reine angew. Math. 232 77-101 (1968). | MR | Zbl

[3] Rohrbach, H.: Ein Beitrag zur additiven Zahlentheorie. Math. Z. 42 1-30 (1937). | DOI | MR

[4] Stanton, R. G., Bate, J. A., Mullin, R. C.: Some tables for the postage stamp problem. Congr. Numer., Proceedings of the Fourth Manitoba Conference on Numerical Mathematics, Winnipeg 12 351-356 (1974). | MR

[5] Stöhr, A.: Gelöste and ungelöste Fragen über Basen der natürlichen Zahlenreihe, I. J. reine Angew. Math. 194 40-65 (1955). | MR

[6] Stöhr, A.: Gelöste and ungelöste Fragen über Basen der natürlichen Zahlenreihe, II. J. reine Angew. Math. 194 111-140 (1955). | MR