A036241 a(1)=1, a(2)=2, a(3)=3; for n >= 3, a(n) is smallest number such that all a(i) for 1 <= i <= n are distinct, all a(i)+a(j) for 1 <= i < j <= n are distinct and all a(i)+a(j)+a(k) for 1 <= i < j < k <= n are distinct.

1, 2, 3, 5, 8, 14, 25, 45, 82, 140, 235, 388, 559, 839, 1286, 1582, 2221, 3144, 4071, 5795, 6872, 9204, 11524, 13796, 17686, 21489, 26019, 31080, 37742, 45067, 53144, 58365, 67917, 78484, 91767, 106513, 118600, 133486, 147633, 166034, 174717

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• nice

Examples

			For {1,2,3,4} we have 1+4 = 2+3, so a(4) is not 4. For {1,2,3,5} the terms 1, 2, 3, 5 are distinct, the sums 1+2, 1+3, 1+5, 2+3, 2+5, 3+5 are distinct and the sums 1+2+3, 1+2+5, 1+3+5, 2+3+5 are distinct, so a(4) = 5.
		

References

• Letter from V. Jooste, Pretoria, South Africa, Sep. 8, 1975.

Links

• Chai Wah Wu, Table of n, a(n) for n = 1..237 (n = 1..75 from Reinhard Zumkeller)

Crossrefs

Cf. A011185, A062065, A051912, A060276.

Programs

Haskell

import qualified Data.Set as Set (null, map)
import Data.Set (empty, fromList, toList, intersect, union)
a036241 n = a036241_list !! (n-1)
a036241_list = f [1..] [] empty empty where
   f (x:xs) ys s2 s3
    | null (s2' `intersect` y2s) && null (s3' `intersect` y3s)
      = x : f xs (x:ys) (fromList s2' `union` s2) (fromList s3' `union` s3)
    | otherwise = f xs ys s2 s3
    where s2' = sort $ map (x +) ys
          s3' = sort $ map (x +) y2s
          y2s = toList s2
          y3s = toList s3
-- Reinhard Zumkeller, Oct 02 2011

Mathematica

a[1]=1; a[2]=2; a[3]=3; a[n_] := a[n] = Catch[For[an = a[n-1] + 1, True, an++, a[n] = an; t2 = Flatten[Table[a[i] + a[j], {i, 1, n}, {j, i+1, n}]]; If[n*(n-1)/2 == Length[Union[t2]], t3 = Flatten[Table[a[i] + a[j] + a[k], {i, 1, n}, {j, i+1, n}, {k, j+1, n}]]; If[ n*(n-1)*(n-2)/6 == Length[Union[t3]], Throw[an]]]]]; Table[Print[a[n]]; a[n], {n, 1, 41}] (* Jean-François Alcover, Jul 24 2012 *)

PARI

{unique(v)=local(b); b=1; for(j=2,length(v),if(v[j-1]==v[j],b=0)); b}
{newsort(u,v,q)=local(s); s=[]; for(i=1,length(v),s=concat(s,v[i]+q)); vecsort(concat(u,s))}
{m=175000; print1(1,",",2,",",3,","); w1=[1,2,3]; w2=[3,4,5]; w3=[6]; q=4; while(q
				

Python

from itertools import count, islice
def A036241_gen(): # generator of terms
    aset2, aset3 = {3,4,5}, {6}
    yield from (alist:=[1,2,3])
    for k in count(4):
        bset2, bset3 = set(), set()
        for a in alist:
            if (b2:=a+k) in aset2:
                break
            bset2.add(b2)
        else:
            for a2 in aset2:
                if (b3:=a2+k) in aset3:
                    break
                bset3.add(b3)
            else:
                yield k
                alist.append(k)
                aset2.update(bset2)
                aset3.update(bset3)
A036241_list = list(islice(A036241_gen(),20)) # Chai Wah Wu, Sep 10 2023

Extensions

Better description and more terms from Naohiro Nomoto, Jul 02 2001

Edited by and terms a(30) to a(41) from Klaus Brockhaus, May 21 2003