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A383855 The n-th term of the sequence is k after every k*(k+1)/2 occurrences of 1, with multiple values following a 1 listed in order.

%I A383855 #16 May 24 2025 16:21:15
%S A383855 1,1,1,2,1,1,1,2,3,1,1,1,2,1,4,1,1,2,3,1,1,1,2,5,1,1,1,2,3,1,1,4,1,2,
%T A383855 6,1,1,1,2,3,1,1,1,2,1,7,1,1,2,3,4,5,1,1,1,2,1,1,1,2,3,8,1,1,1,2,1,4,
%U A383855 1,1,2,3,6,1,1,1,2,5,9,1,1,1,2,3,1,1,4,1,2,1,1,1,2,3,1,10
%N A383855 The n-th term of the sequence is k after every k*(k+1)/2 occurrences of 1, with multiple values following a 1 listed in order.
%C A383855 The frequencies of the terms follow the Yule-Simon distribution with parameter value 1. The geometric mean approaches A245254 in the limit.
%H A383855 Jwalin Bhatt, <a href="/A383855/b383855.txt">Table of n, a(n) for n = 1..10000</a>
%H A383855 Wikipedia, <a href="https://en.wikipedia.org/wiki/Yule%E2%80%93Simon_distribution">Yule-Simon distribution</a>
%e A383855 After every ((2*3)/2=3) ones we see a 2,
%e A383855 after every ((3*4)/2=6) ones we see a 3,
%e A383855 after every ((4*5)/2=10) ones we see a 4 and so on.
%o A383855 (Python)
%o A383855 from itertools import islice
%o A383855 def beta_distribution_generator():
%o A383855     num_ones, num_reached = 0, 1
%o A383855     while num_ones := num_ones+1:
%o A383855         yield 1
%o A383855         for num in range(2, num_reached+2):
%o A383855             if num_ones % (num*(num+1)//2) == 0:
%o A383855                 yield num
%o A383855                 num_reached += num == num_reached+1
%o A383855 A383855 = list(islice(beta_distribution_generator(), 120))
%Y A383855 Cf. A245254, A383899.
%K A383855 nonn
%O A383855 1,4
%A A383855 _Jwalin Bhatt_, May 12 2025