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单词 MianChowlaSequence
释义

Mian-Chowla sequence


The Mian-Chowla sequenceMathworldPlanetmath is a B2 sequence with a1=1 and an for n>2 being the smallest integer such that each pairwise sum ai+aj is distinct, where 0<i<(n+1) and likewise for j, that is, 1ijn. The case i=j is always considered.

At the beginning, with a1, there is only one pairwise sum, 2. a2 can be 2 since the pairwise sums then are 2, 3 and 4. a3 can’t be 3 because then there would be the pairwise sums 1 + 3 = 2 + 2 = 4. Thus a3=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 a1=0, the resulting sequence is the same except each term is one less.

Rachel Lewis noticed that

i=11ai2.1585

, 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 TheoryMathworldPlanetmathPlanetmath, New York: Springer (2003)
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更新时间:2025/5/4 13:27:38