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A382093 Sequence where k is appended after every (k-1)! occurrences of 1, with multiple values following a 1 listed in order.

%I A382093 #29 May 24 2025 16:21:35
%S A382093 1,2,1,2,3,1,2,1,2,3,1,2,1,2,3,4,1,2,1,2,3,1,2,1,2,3,1,2,1,2,3,4,1,2,
%T A382093 1,2,3,1,2,1,2,3,1,2,1,2,3,4,1,2,1,2,3,1,2,1,2,3,1,2,1,2,3,4,5,1,2,1,
%U A382093 2,3,1,2,1,2,3,1,2,1,2,3,4,1,2,1,2,3,1,2,1,2,3,1,2,1,2,3,4,1,2,1,2,3,1,2,1,2,3,1,2,1,2,3,4,1,2,1,2,3,1,2
%N A382093 Sequence where k is appended after every (k-1)! occurrences of 1, with multiple values following a 1 listed in order.
%C A382093 The frequencies of the terms follow the Poisson distribution with parameter value 1.
%C A382093 The geometric mean approaches A382095 in the limit. In general, for parameter value p it approaches Product_{k>=2} k^(((p^(k-1))*(e^(-p)))/(k-1)!).
%H A382093 Jwalin Bhatt, <a href="/A382093/b382093.txt">Table of n, a(n) for n = 1..10000</a>
%H A382093 Wikipedia, <a href="https://en.wikipedia.org/wiki/Poisson_distribution">Poisson distribution</a>
%e A382093 Every 1 is followed by a 2 because (2-1)! = 1,
%e A382093 after every (2!=2) ones we see a 3,
%e A382093 after every (3!=6) ones we see a 4 and so on.
%o A382093 (Python)
%o A382093 from itertools import islice
%o A382093 from math import factorial
%o A382093 def poisson_distribution_generator():
%o A382093     num_ones, num_reached = 0, 1
%o A382093     while num_ones := num_ones+1:
%o A382093         yield 1
%o A382093         for num in range(2, num_reached+2):
%o A382093             if num_ones % factorial(num-1) == 0:
%o A382093                 yield num
%o A382093                 num_reached += num == num_reached+1
%o A382093 A382093 = list(islice(poisson_distribution_generator(), 120))
%Y A382093 Cf. A381522, A381900, A382095.
%K A382093 nonn
%O A382093 1,2
%A A382093 _Jwalin Bhatt_, Mar 25 2025