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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 A376942 #22 Nov 08 2024 08:38:02
%S A376942 1,1,1,2,5,100,1,1,1,1,3,53,4947,66072132,1,1,1,1,1,1,23,5270,
%T A376942 27999510,1,1,1,1,1,1,1,2,4,28,8851,1395426533,3665346274452116372,
%U A376942 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 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.
%C A376942 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 A376942 Conjecture: all rows have finite length.
%H A376942 Scott R. Shannon, <a href="/A376942/b376942.txt">Table of n, a(n) for n = 1..797</a>
%H A376942 Scott R. Shannon, <a href="/A376942/a376942.txt">Unflattened table for n = 1..25</a>.
%e A376942 row(1) = 1 as 1/(1*1) = 1.
%e A376942 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 A376942 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 A376942 .
%e A376942 The table begins:
%e A376942 1;
%e A376942 1, 1, 2, 5, 100;
%e A376942 1, 1, 1, 1, 3, 53, 4947, 66072132;
%e A376942 1, 1, 1, 1, 1, 1, 23, 5270, 27999510;
%e A376942 1, 1, 1, 1, 1, 1, 1, 2, 4, 28, 8851, 1395426533, 3665346274452116372, 53925647181443925794153448868309082440;
%e A376942 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 7, 95, 54570, 3932969040;
%e A376942 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 6, 45, 2685, 8685204, 98388241169400;
%e A376942 .
%e A376942 .
%e A376942 .
%e A376942 See the attached file for rows up to n = 25.
%Y A376942 Cf. A001008, A002805, A002387, A375781, A376056, A269993.
%K A376942 nonn,tabf
%O A376942 1,4
%A A376942 _Scott R. Shannon_, Oct 12 2024