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A376062
Lexicographically earliest sequence of positive integers a(1), a(2), a(3), ... such that for any n > 0, S(n) = Sum_{k = 1..n} b(k)/a(k) < 1, where {b(k)} is the sequence {7/6, 5/4, 5/4, 5/4, ...}.
5
2, 4, 13, 157, 24493, 599882557, 359859081592975693, 129498558604939936868397356895854557, 16769876680757063368089314196389622249367851612542961252860614401811693
OFFSET
1,1
COMMENTS
This sequence and A376186 were discovered by Rémy Sigrist on Sep 09 2024. The two sequences {b(1)=7/6, b(k)=5/4 for k>1} and {b(1)=5/4, b(2*k)=3/2, b(2*k+1)=6/5 for k>0} are the first sequences {b(i)} discovered with the property that the sums S(n) do not converge to numbers of the form (e_n - 1)/e_n as n-> oo.
This is essentially the same sequence as A004168 and A082732.
LINKS
N. J. A. Sloane, A Nasty Surprise in a Sequence and Other OEIS Stories, Experimental Mathematics Seminar, Rutgers University, Oct 10 2024, Youtube video; Slides [Mentions this sequence]
FORMULA
a(n+1) = a(n)^2 - a(n) + 1 for n >= 2.
MATHEMATICA
Join[{2}, RecurrenceTable[{a[n+1] == a[n]^2 - a[n] + 1, a[2] == 4}, a, {n, 2, 9}]] (* Amiram Eldar, Sep 15 2024 *)
KEYWORD
nonn
AUTHOR
N. J. A. Sloane, Sep 14 2024
STATUS
approved