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A005282 Mian-Chowla sequence (a B_2 sequence): a(1) = 1; for n>1, a(n) = smallest number > a(n-1) such that the pairwise sums of elements are all distinct.
(Formerly M1094)
51

%I M1094

%S 1,2,4,8,13,21,31,45,66,81,97,123,148,182,204,252,290,361,401,475,565,

%T 593,662,775,822,916,970,1016,1159,1312,1395,1523,1572,1821,1896,2029,

%U 2254,2379,2510,2780,2925,3155,3354,3591,3797,3998,4297,4433,4779,4851

%N Mian-Chowla sequence (a B_2 sequence): a(1) = 1; for n>1, a(n) = smallest number > a(n-1) such that the pairwise sums of elements are all distinct.

%C An alternative definition is to start with 1 and then continue with the least number such that all pairwise differences of distinct elements are all distinct. - _Jens Voß_, Feb 04 2003. [However, compare A003022 and A227590. - _N. J. A. Sloane_, Apr 08 2016]

%C R. Lewis points out, at the first Weisstein link, that S, the sum of the reciprocals of this sequence, satisfies 2.158435 <= S <= 2.158677. Similarly, the sum of the squares of reciprocals of this sequence converges to approximately 1.33853369 and the sum of the cube of reciprocals of this sequence converges to approximately 1.14319352. - _Jonathan Vos Post_, Nov 21 2004

%C Let S denote the reciprocal sum of a(n). Then 2.158452685 <= S <= 2.158532684. - _Raffaele Salvia_, Jul 19 2014

%C From _Thomas Ordowski_, Sep 19 2014: (Start)

%C Known estimate: n^2/2 + O(n) < a(n) < n^3/6 + O(n^2).

%C Conjecture: a(n) ~ n^3 / log(n)^2.

%C (End)

%D S. R. Finch, Mathematical Constants, Cambridge, 2003, Section 2.20.2.

%D R. K. Guy, Unsolved Problems in Number Theory, E28.

%D A. M. Mian and S. D. Chowla, On the B_2-sequences of Sidon, Proc. Nat. Acad. Sci. India, A14 (1944), 3-4.

%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

%H T. D. Noe, <a href="/A005282/b005282.txt">Table of n, a(n) for n=1..5818</a> (terms less than 2*10^9)

%H Raffaele Salvia, <a href="/A005282/a005282.txt">Table of n, a(n) for n=1...25000</a>

%H R. Salvia, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL18/Salvia/salvia3.html">A New Lower Bound for the Distinct Distance Constant</a>, J. Int. Seq. 18 (2015) # 15.4.8.

%H N. J. A. Sloane, <a href="/A001149/a001149.pdf">Handwritten notes on Self-Generating Sequences, 1970</a> (note that A1148 has now become A005282)

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/B2-Sequence.html">B2 Sequence.</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Mian-ChowlaSequence.html">Chowla Sequence.</a>

%H Zhang Zhen-Xiang, <a href="http://dx.doi.org/10.1090/S0025-5718-1993-1181334-7">A B_2-sequence with larger reciprocal sum</a>, Math. Comp. 60 (1993), 835-839.

%H <a href="/index/Br#B_2">Index entries for B_2 sequences.</a>

%F a(n) = A025582(n) + 1.

%F a(n) = (A034757(n)+1)/2.

%e The second term is 2 because the 3 pairwise sums 1+1=2, 1+2=3, 2+2=4 are all distinct.

%e The third term cannot be 3 because 1+3 = 2+2. But it can be 4, since 1+4=5, 2+4=6, 4+4=8 are distinct and distinct from the earler sums 1+1=2, 1+2=3, 2+2=4.

%p a[1]:= 1: P:= {2}: A:= {1}:

%p for n from 2 to 100 do

%p for t from a[n-1]+1 do

%p Pt:= map(`+`,A union {t},t);

%p if Pt intersect P = {} then break fi

%p od:

%p a[n]:= t;

%p A:= A union {t};

%p P:= P union Pt;

%p od:

%p seq(a[n],n=1..100); # _Robert Israel_, Sep 21 2014

%t t = {1}; sms = {2}; k = 1; Do[k++; While[Intersection[sms, k + t] != {}, k++]; sms = Join[sms, t + k, {2 k}]; AppendTo[t, k], {49}]; t (* _T. D. Noe_, Mar 02 2011 *)

%o (Haskell)

%o import Data.Set (Set, empty, insert, member)

%o a005282 n = a005282_list !! (n-1)

%o a005282_list = sMianChowla [] 1 empty where

%o sMianChowla :: [Integer] -> Integer -> Set Integer -> [Integer]

%o sMianChowla sums z s | s' == empty = sMianChowla sums (z+1) s

%o | otherwise = z : sMianChowla (z:sums) (z+1) s

%o where s' = try (z:sums) s

%o try :: [Integer] -> Set Integer -> Set Integer

%o try [] s = s

%o try (x:sums) s | (z+x) `member` s = empty

%o | otherwise = try sums $ insert (z+x) s

%o -- _Reinhard Zumkeller_, Mar 02 2011

%Y Cf. A051788, A080200 (for differences between terms).

%Y Different from A046185. Cf. A011185.

%Y See also A003022, A227590.

%Y A259964 has a greater sum of reciprocals.

%K nonn,nice

%O 1,2

%A _N. J. A. Sloane_ and _Simon Plouffe_

%E Examples added by _N. J. A. Sloane_, Jun 01 2008

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Last modified June 19 17:22 EDT 2018. Contains 305594 sequences. (Running on oeis4.)