Mian-Chowla sequence
The Mian-Chowla sequence^{} 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 $$ and likewise for $j$, that is, $1\le i\le j\le n$. The case $i=j$ is always considered.
At the beginning, with ${a}_{1}$, there is only one pairwise sum, 2. ${a}_{2}$ can be 2 since the pairwise sums then are 2, 3 and 4. ${a}_{3}$ can’t be 3 because then there would be the pairwise sums 1 + 3 = 2 + 2 = 4. Thus ${a}_{3}=4$. The sequence, listed in A005282 of Sloane’s OEIS, continues 8, 13, 21, 31, 45, 66, 81, 97, 123, 148, 182, 204, 252, 290, 361, 401, 475, … If we define ${a}_{1}=0$, the resulting sequence is the same except each term is one less.
Rachel Lewis noticed that
$$\sum _{i=1}^{\mathrm{\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 R. K. Guy Unsolved Problems in Number Theory^{}, New York: Springer (2003)
Title | Mian-Chowla sequence |
---|---|
Canonical name | MianChowlaSequence |
Date of creation | 2013-03-22 16:27:49 |
Last modified on | 2013-03-22 16:27:49 |
Owner | PrimeFan (13766) |
Last modified by | PrimeFan (13766) |
Numerical id | 6 |
Author | PrimeFan (13766) |
Entry type | Definition |
Classification | msc 11B13 |