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Mian–Chowla sequence

Sequence of numbers with distinct sums

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 n1, 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, ... .

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, ... ).

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)

Info: Wikipedia Source

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integer-sequences

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