%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 MianChowla sequence (a B_2 sequence): a(1) = 1; for n>1, a(n) = smallest number > a(n1) 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_2sequences of Sidon, Proc. Nat. Acad. Sci. India, A14 (1944), 34.
%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 SelfGenerating Sequences, 1970</a> (note that A1148 has now become A005282)
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/B2Sequence.html">B2 Sequence.</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/MianChowlaSequence.html">Chowla Sequence.</a>
%H Zhang ZhenXiang, <a href="http://dx.doi.org/10.1090/S00255718199311813347">A B_2sequence with larger reciprocal sum</a>, Math. Comp. 60 (1993), 835839.
%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[n1]+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 !! (n1)
%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
