Mian–Chowla sequence

In mathematics, the Mian–Chowla sequence is an integer sequence defined

recursively in the following way. The sequence starts with

: $a_1 = 1.$

Then for $n>1$ , $a_n$ is the smallest integer such that every pairwise sum

: $a_i + a_j$

is distinct, for all $i$ and $j$ less than or equal to $n$ .

Properties

Initially, with $a_1$ , there is only one pairwise sum, 1 + 1 = 2. The next term in the sequence, $a_2$ , is 2 since the pairwise sums then are 2, 3 and 4, i.e., they are distinct. Then, $a_3$ can't be 3 because there would be the non-distinct pairwise sums 1 + 3 = 2 + 2 = 4. We find then that $a_3 = 4$ , with the pairwise sums being 2, 3, 4, 5, 6 and 8. The sequence thus begins

:1, 2, 4, 8, 13, 21, 31, 45, 66, 81, 97, 123, 148, 182, 204, 252, 290, 361, 401, 475, ... {{OEIS|id=A005282}}.

Similar sequences

If we define $a_1 = 0$ , the resulting sequence is the same except each term is one less (that is, 0, 1, 3, 7, 12, 20, 30, 44, 65, 80, 96, ... {{OEIS2C|id=A025582}}).

History

The sequence was invented by Abdul Majid Mian and Sarvadaman Chowla.

References

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

Category:Integer sequences