

A005282


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.
(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
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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
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_2sequences of Sidon, Proc. Nat. Acad. Sci. India, A14 (1944), 34.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).


LINKS



FORMULA



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[n1]+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:


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 !! (n1)
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(n1, k, A[n]A[nk]))={ 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[k1..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


CROSSREFS

A259964 has a greater sum of reciprocals.


KEYWORD

nonn,nice


AUTHOR



EXTENSIONS



STATUS

approved



