Mian-Chowla sequence
The Mian-Chowla sequence is a sequence with and for being the smallest integer such that each pairwise sum is distinct, where and likewise for , that is, . The case is always considered.
At the beginning, with , there is only one pairwise sum, 2. can be 2 since the pairwise sums then are 2, 3 and 4. can’t be 3 because then there would be the pairwise sums 1 + 3 = 2 + 2 = 4. Thus . 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 , the resulting sequence is the same except each term is one less.
Rachel Lewis noticed that
, 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 R. K. Guy Unsolved Problems in Number Theory
, New York: Springer (2003)