A334126 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.

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, 29, 24, 27, 32, 25, 33, 34, 35, 28, 46, 31, 37, 47, 38, 39, 49, 41, 48, 40, 42, 50, 43, 44, 51, 45, 52, 53, 56, 54, 58, 59, 55, 62, 65, 68, 67, 57, 63, 64, 70, 61, 69

Offset: 1

Author

Jinyuan Wang, May 10 2020

Keywords

• nonn

Comments

I conjecture that every number eventually appears.

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.

Crossrefs

Cf. A001597, A127397.

Programs

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; }