Mian–Chowla sequence

In mathematics, the Mian–Chowla sequence is an integer sequence defined recursively in the following way. The sequence starts with

a₁ = 1.

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

a_i + a_j

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

Initially, with a₁, there is only one pairwise sum, 1 + 1 = 2. The next term in the sequence, a₂, is 2 since the pairwise sums then are 2, 3 and 4, i.e., they are distinct. Then, a₃ can't be 3 because there would be the non-distinct pairwise sums 1 + 3 = 2 + 2 = 4. We find then that a₃ = 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, ... (sequence A005282 in OEIS).

If we define a₁ = 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, ...  A025582).

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)

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