A364132 - OEIS

%I #23 Aug 06 2023 08:18:48

%S 1,2,4,5,7,10,12,13,15,18,21,24,25,29,30,33,36,38,41,47,50,52

%N a(n) is the smallest positive integer such that from the set {1, 2, ..., a(n)} one can choose an increasing sequence (s(1), s(2), ..., s(n)) in which every segment has a unique sum of elements.

%C A segment is a subsequence of consecutive elements.

%e a(6) = 10, because there exists a 6-element increasing sequence on {1,2,...,10} with unique segment sums, namely (1,2,4,5,8,10) and 10 is the least positive integer with that property. The sums in the segments are: 1, 2, 4, 5, 8, 10 for 1-element segments; 3, 6, 9, 13, 18 for 2-element segments; 7, 11, 17, 23 for 3-element segments; 12, 19, 27 for 4-element segments; 20, 29 for 5-element segments; and 30 for the full set.

%e a(13) = 25 and the corresponding 13-element subsequence is (1,2,11,15,16,17,18,19,20,21,22,24,25).

%o (PARI) a(n, m=2*n) = my(k=1, s=vector(n, i, []), t, u=m, v=vector(n)); while(k>1||v[1]<u-n, t=0; v[k]++; if(k==n, if(v[n]<u, if(!#setintersect(vector(n, i, t=t+v[n+1-i]), s[n]), u=v[n]), k--), if(v[k]<u+k-n, s[k+1]=setunion(vector(k, i, t=t+v[k+1-i]), s[k]); if(#s[k+1]==k*(k+1)/2, v[k+1]=v[k]; k++), k--))); if(u<m, u, a(n, m+4)); \\ _Jinyuan Wang_, Jul 10 2023

%Y Cf. A364153 (without monotonicity assumption).

%Y Cf. A276661, A363446.

%K nonn,hard,more

%O 1,2

%A _Bartlomiej Pawlik_, Jul 10 2023

%E a(14)-a(22) from _Jinyuan Wang_, Jul 10 2023