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A005282
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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)
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59
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1, 2, 4, 8, 13, 21, 31, 45, 66, 81, 97, 123, 148, 182, 204, 252, 290, 361, 401, 475, 565, 593, 662, 775, 822, 916, 970, 1016, 1159, 1312, 1395, 1523, 1572, 1821, 1896, 2029, 2254, 2379, 2510, 2780, 2925, 3155, 3354, 3591, 3797, 3998, 4297, 4433, 4779, 4851
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OFFSET
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1,2
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COMMENTS
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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]
Rachel Lewis points out [see 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; comment changed by N. J. A. Sloane, Jan 02 2020
Let S denote the reciprocal sum of a(n). Then 2.158452685 <= S <= 2.158532684. - Raffaele Salvia, Jul 19 2014
Known estimate: n^2/2 + O(n) < a(n) < n^3/6 + O(n^2).
Conjecture: a(n) ~ n^3 / log(n)^2.
(End)
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REFERENCES
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S. R. Finch, Mathematical Constants, Cambridge, 2003, Section 2.20.2.
R. K. Guy, Unsolved Problems in Number Theory, E28.
A. M. Mian and S. D. Chowla, On the B_2-sequences of Sidon, Proc. Nat. Acad. Sci. India, A14 (1944), 3-4.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
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FORMULA
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EXAMPLE
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The second term is 2 because the 3 pairwise sums 1+1=2, 1+2=3, 2+2=4 are all distinct.
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 earlier sums 1+1=2, 1+2=3, 2+2=4.
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MAPLE
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a[1]:= 1: P:= {2}: A:= {1}:
for n from 2 to 100 do
for t from a[n-1]+1 do
Pt:= map(`+`, A union {t}, t);
if Pt intersect P = {} then break fi
od:
a[n]:= t;
A:= A union {t};
P:= P union Pt;
od:
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MATHEMATICA
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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 *)
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PROG
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(Haskell)
import Data.Set (Set, empty, insert, member)
a005282 n = a005282_list !! (n-1)
a005282_list = sMianChowla [] 1 empty where
sMianChowla :: [Integer] -> Integer -> Set Integer -> [Integer]
sMianChowla sums z s | s' == empty = sMianChowla sums (z+1) s
| otherwise = z : sMianChowla (z:sums) (z+1) s
where s' = try (z:sums) s
try :: [Integer] -> Set Integer -> Set Integer
try [] s = s
try (x:sums) s | (z+x) `member` s = empty
| otherwise = try sums $ insert (z+x) s
(PARI) A005282_vec(N, A=[1], U=[0], D(A, n=#A)=vector(n-1, k, A[n]-A[n-k]))={ while(#A<N, my(k=1); A=concat(A, A[#A]+U[1]); until(!setintersect(U, D(A)), A[#A]++); U=setunion(U, D(A)); while(k<#U && U[k++]<U[1]+k, ); k>2 && U=U[k-1..-1]); A} \\ M. F. Hasler, Oct 09 2019
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CROSSREFS
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A259964 has a greater sum of reciprocals.
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KEYWORD
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nonn,nice
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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