A376942 - OEIS

A376942

Irregular table read by rows: row(n) is the lexicographically earliest sequence of positive integers a(n,1), a(n,2), ... a(n,k) such that Sum_{m = n..(n+k-1)} 1/(m*a(n,m-n+1)) <= 1.

%I #22 Nov 08 2024 08:38:02

%S 1,1,1,2,5,100,1,1,1,1,3,53,4947,66072132,1,1,1,1,1,1,23,5270,

%T 27999510,1,1,1,1,1,1,1,2,4,28,8851,1395426533,3665346274452116372,

%U 53925647181443925794153448868309082440,1,1,1,1,1,1,1,1,1,3,7,95,54570,3932969040,1,1,1,1,1,1,1,1,1,1,1,6,45,2685,8685204,98388241169400

%N Irregular table read by rows: row(n) is the lexicographically earliest sequence of positive integers a(n,1), a(n,2), ... a(n,k) such that Sum_{m = n..(n+k-1)} 1/(m*a(n,m-n+1)) <= 1.

%C The terms in each row can grow rapidly in size, e.g., the 63rd and final term in row(25), 36333...86400, has 1728101 digits.

%C Conjecture: all rows have finite length.

%H Scott R. Shannon, <a href="/A376942/b376942.txt">Table of n, a(n) for n = 1..797</a>

%H Scott R. Shannon, <a href="/A376942/a376942.txt">Unflattened table for n = 1..25</a>.

%e row(1) = 1 as 1/(1*1) = 1.

%e row(2) = 1, 1, 2, 5, 100 as 1/(2*1) + 1/(3*1) + 1/(4*2) + 1/(5*5) + 1/(6*100) = 1.

%e row(3) = 1, 1, 1, 1, 3, 53, 4947, 66072132 as 1/(3*1) + 1/(4*1) + 1/(5*1) + 1/(6*1) + 1/(7*3) + 1/(8*53) + 1/(9*4947) + 1/(10*66072132) = 1.

%e .

%e The table begins:

%e 1;

%e 1, 1, 2, 5, 100;

%e 1, 1, 1, 1, 3, 53, 4947, 66072132;

%e 1, 1, 1, 1, 1, 1, 23, 5270, 27999510;

%e 1, 1, 1, 1, 1, 1, 1, 2, 4, 28, 8851, 1395426533, 3665346274452116372, 53925647181443925794153448868309082440;

%e 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 7, 95, 54570, 3932969040;

%e 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 6, 45, 2685, 8685204, 98388241169400;

%e .

%e See the attached file for rows up to n = 25.

%Y Cf. A001008, A002805, A002387, A375781, A376056, A269993.

%K nonn,tabf

%O 1,4

%A _Scott R. Shannon_, Oct 12 2024