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%I #17 Jun 22 2022 09:25:17
%S 1,2,3,7,5,6,17,12,8,10,4,16,13,11,15,18,19,9,14,23,20,30,22,21,26,36,
%T 29,24,27,32,25,33,34,35,28,46,31,37,47,38,39,49,41,48,40,42,50,43,44,
%U 51,45,52,53,56,54,58,59,55,62,65,68,67,57,63,64,70,61,69
%N a(n) is the smallest positive integer not already in the sequence such that Sum_{i=k..n} a(i) is not a perfect power for 0 < k < n; start with a(1)=1.
%C I conjecture that every number eventually appears.
%C Let b(1) = 1; b(2*m) is the least positive integer not occurring earlier in b(i), i=1..2*m-2; b(2*m-1) is the least positive integer not already in {b(n)} such that Sum_{i=j..2*m-1} b(i) and Sum_{i=k..2*m} b(i) are not perfect powers for 0 < j < 2*m-1 and 0 < k < 2*m. Then {b(n)} is a permutation of the positive integers such that Sum_{i=k..m} b(i) is not a perfect power for any 0 < k < m.
%o (PARI) lista(nn) = {my(k, s, v=vector(nn)); v[1]=1; for(n=2, nn, k=s=2; while(vecsearch(vecsort(v), k) || sum(i=1, n-1, ispower(s+=v[n-i])), s=k++); v[n]=k); v; }
%Y Cf. A001597, A127397.
%K nonn
%O 1,2
%A _Jinyuan Wang_, May 10 2020
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