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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)
60
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. [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
From Thomas Ordowski, Sep 19 2014: (Start)
Known estimate: n^2/2 + O(n) < a(n) < n^3/6 + O(n^2).
Conjecture: a(n) ~ n^3 / log(n)^2.
(End)
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).
LINKS
T. D. Noe, Table of n, a(n) for n=1..5818 (terms less than 2*10^9)
Thomas Bloom, Problem 340, Erdős Problems.
Rachel Lewis, Mian-Chowla and B2 sequences, 1999. [Thanks to Steven Finch for providing this document. Included with permission. - N. J. A. Sloane, Jan 02 2020]
Kevin O'Bryant, B_h-Sets and Rigidity, arXiv:2312.10910 [math.NT], 2023.
R. Salvia, A New Lower Bound for the Distinct Distance Constant, J. Int. Seq. 18 (2015) # 15.4.8.
N. J. A. Sloane, Handwritten notes on Self-Generating Sequences, 1970 (note that A1148 has now become A005282)
Eric Weisstein's World of Mathematics, B2 Sequence.
Eric Weisstein's World of Mathematics, Chowla Sequence.
Zhang Zhen-Xiang, A B_2-sequence with larger reciprocal sum, Math. Comp. 60 (1993), 835-839.
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 earlier sums 1+1=2, 1+2=3, 2+2=4.
MAPLE
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:
seq(a[n], n=1..100); # Robert Israel, Sep 21 2014
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
(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
(Python)
from itertools import count, islice
def A005282_gen(): # generator of terms
aset1, aset2, alist = set(), set(), []
for k in count(1):
bset2 = {k<<1}
if (k<<1) not in aset2:
for d in aset1:
if (m:=d+k) in aset2:
break
bset2.add(m)
else:
yield k
alist.append(k)
aset1.add(k)
aset2 |= bset2
A005282_list = list(islice(A005282_gen(), 30)) # Chai Wah Wu, Sep 05 2023
CROSSREFS
Row 2 of A347570.
Cf. A051788, A080200 (for differences between terms).
Different from A046185. Cf. A011185.
See also A003022, A227590.
A259964 has a greater sum of reciprocals.
Sequence in context: A292774 A026039 A004978 * A046185 A259964 A218913
KEYWORD
nonn,nice
AUTHOR
EXTENSIONS
Examples added by N. J. A. Sloane, Jun 01 2008
STATUS
approved

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Last modified July 12 22:00 EDT 2024. Contains 374257 sequences. (Running on oeis4.)