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# Mian-Chowla sequence

The Mian-Chowla sequence is a $B_{2}$ sequence with $a_{1}=1$ and $a_{n}$ for $n>2$ being the smallest integer such that each pairwise sum $a_{i}+a_{j}$ is distinct, where $0<i<(n+1)$ and likewise for $j$, that is, $1\leq i\leq j\leq n$. The case $i=j$ is always considered.

At the beginning, with $a_{1}$, there is only one pairwise sum, 2. $a_{2}$ can be 2 since the pairwise sums then are 2, 3 and 4. $a_{3}$ can’t be 3 because then there would be the pairwise sums 1 + 3 = 2 + 2 = 4. Thus $a_{3}=4$. The sequence, listed in A005282 of Sloane’s OEIS, continues 8, 13, 21, 31, 45, 66, 81, 97, 123, 148, 182, 204, 252, 290, 361, 401, 475, … If we define $a_{1}=0$, the resulting sequence is the same except each term is one less.

Rachel Lewis noticed that

$\sum_{{i=1}}^{\infty}\frac{1}{a_{i}}\equiv 2.1585$ |

, a constant listed in Finch’s book.

One way to calculate the Mian-Chowla sequence in Mathematica is thus:

a = Table[1, {40}]; n = 2; test = 1; While[n < 41, mcFlag = False; While[Not[mcFlag], test++; a[[n]] = test; pairSums = Flatten[Table[a[[i]] + a[[j]], {i, n}, {j, i, n}]]; mcFlag = TrueQ[Length[pairSums] == Length[Union[pairSums]]] ]; n++ ]; a

# References

- 1 S. R. Finch, Mathematical Constants, Cambridge (2003): Section 2.20.2
- 2 R. K. Guy Unsolved Problems in Number Theory, New York: Springer (2003)

## Mathematics Subject Classification

11B13*no label found*

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