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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.

%I A036241 #33 Sep 11 2023 17:50:57
%S A036241 1,2,3,5,8,14,25,45,82,140,235,388,559,839,1286,1582,2221,3144,4071,
%T A036241 5795,6872,9204,11524,13796,17686,21489,26019,31080,37742,45067,53144,
%U A036241 58365,67917,78484,91767,106513,118600,133486,147633,166034,174717
%N 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.
%D A036241 Letter from V. Jooste, Pretoria, South Africa, Sep. 8, 1975.
%H A036241 Chai Wah Wu, <a href="/A036241/b036241.txt">Table of n, a(n) for n = 1..237</a> (n = 1..75 from Reinhard Zumkeller)
%e A036241 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.
%t A036241 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 *)
%o A036241 (PARI) {unique(v)=local(b); b=1; for(j=2,length(v),if(v[j-1]==v[j],b=0)); b}
%o A036241 {newsort(u,v,q)=local(s); s=[]; for(i=1,length(v),s=concat(s,v[i]+q)); vecsort(concat(u,s))}
%o A036241 {m=175000; print1(1,",",2,",",3,","); w1=[1,2,3]; w2=[3,4,5]; w3=[6]; q=4; while(q<m,y1=concat(w1,q); y2=newsort(w2,w1,q); y3=newsort(w3,w2,q); if(unique(y1)&&unique(y2)&&unique(y3),w1=y1; w2=y2; w3=y3; print1(q,",")); q=q+1)}
%o A036241 (Haskell)
%o A036241 import qualified Data.Set as Set (null, map)
%o A036241 import Data.Set (empty, fromList, toList, intersect, union)
%o A036241 a036241 n = a036241_list !! (n-1)
%o A036241 a036241_list = f [1..] [] empty empty where
%o A036241    f (x:xs) ys s2 s3
%o A036241     | null (s2' `intersect` y2s) && null (s3' `intersect` y3s)
%o A036241       = x : f xs (x:ys) (fromList s2' `union` s2) (fromList s3' `union` s3)
%o A036241     | otherwise = f xs ys s2 s3
%o A036241     where s2' = sort $ map (x +) ys
%o A036241           s3' = sort $ map (x +) y2s
%o A036241           y2s = toList s2
%o A036241           y3s = toList s3
%o A036241 -- _Reinhard Zumkeller_, Oct 02 2011
%o A036241 (Python)
%o A036241 from itertools import count, islice
%o A036241 def A036241_gen(): # generator of terms
%o A036241     aset2, aset3 = {3,4,5}, {6}
%o A036241     yield from (alist:=[1,2,3])
%o A036241     for k in count(4):
%o A036241         bset2, bset3 = set(), set()
%o A036241         for a in alist:
%o A036241             if (b2:=a+k) in aset2:
%o A036241                 break
%o A036241             bset2.add(b2)
%o A036241         else:
%o A036241             for a2 in aset2:
%o A036241                 if (b3:=a2+k) in aset3:
%o A036241                     break
%o A036241                 bset3.add(b3)
%o A036241             else:
%o A036241                 yield k
%o A036241                 alist.append(k)
%o A036241                 aset2.update(bset2)
%o A036241                 aset3.update(bset3)
%o A036241 A036241_list = list(islice(A036241_gen(),20)) # _Chai Wah Wu_, Sep 10 2023
%Y A036241 Cf. A011185, A062065, A051912, A060276.
%K A036241 nonn,nice
%O A036241 1,2
%A A036241 _N. J. A. Sloane_
%E A036241 Better description and more terms from _Naohiro Nomoto_, Jul 02 2001
%E A036241 Edited by and terms a(30) to a(41) from _Klaus Brockhaus_, May 21 2003