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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, $1 \le i \le j \le n$ . 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
- 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)
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