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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)
28
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

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

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

Let S denote the reciprocal sum of a(n). Then 2.158452685 =< S =< 2.158532684. - Raffaele Salvia, Jul 19 2014

REFERENCES

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).

Zhang Zhen-Xiang, A B_2-sequence with larger reciprocal sum, Math. Comp. 60 (1993), 835-839.

LINKS

T. D. Noe, Table of n, a(n) for n=1..5818 (terms less than 2*10^9)

Eric Weisstein's World of Mathematics, B2 Sequence.

Eric Weisstein's World of Mathematics, Chowla Sequence.

Index entries for B_2 sequences.

Raffaele Salvia, Table of n, a(n) for n=1...25000

FORMULA

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

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

EXAMPLE

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 earler sums 1+1=2, 1+2=3, 2+2=4.

MATHEMATICA

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 *)

PROG

(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

-- Reinhard Zumkeller, Mar 02 2011

CROSSREFS

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

Different from A046185. Cf. A011185.

Sequence in context: A115266 A026039 A004978 * A046185 A218913 A241691

Adjacent sequences:  A005279 A005280 A005281 * A005283 A005284 A005285

KEYWORD

nonn,nice

AUTHOR

N. J. A. Sloane and Simon Plouffe

EXTENSIONS

Examples added by N. J. A. Sloane, Jun 01 2008

STATUS

approved

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Last modified September 2 00:06 EDT 2014. Contains 246317 sequences.