# Mian-Chowla sequence

The 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 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
Title Mian-Chowla sequence MianChowlaSequence 2013-03-22 16:27:49 2013-03-22 16:27:49 PrimeFan (13766) PrimeFan (13766) 6 PrimeFan (13766) Definition msc 11B13