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A362372 Inventory of powers. Initialize the sequence with '1'. Then record the number of powers of 1 thus far, then do the same for powers of 2 (2, 4, 8, ...), powers of 3, etc. When the count is zero, do not record a zero; rather start the inventory again with the powers of 1.

%I A362372 #42 May 06 2023 06:58:09
%S A362372 1,1,2,1,3,1,1,5,1,1,7,1,1,9,1,2,10,2,2,10,4,2,1,1,12,6,2,1,1,1,1,16,
%T A362372 8,2,2,1,1,1,1,1,2,21,12,2,2,1,1,1,1,1,2,26,15,2,2,1,1,1,1,1,2,31,18,
%U A362372 2,2,1,1,1,1,1,2,36,21,2,2,1,2,1,1,1
%N A362372 Inventory of powers. Initialize the sequence with '1'. Then record the number of powers of 1 thus far, then do the same for powers of 2 (2, 4, 8, ...), powers of 3, etc. When the count is zero, do not record a zero; rather start the inventory again with the powers of 1.
%C A362372 A variant of the inventory sequence, A342585.
%C A362372 The graph exhibits sharp jumps followed by a rapid decline forming a periodic hockey stick pattern. Larger-scale, near-linear structures also appear.
%C A362372 Periodic patterns in the relative frequency of any given number also are present. For example, perform a rolling count of the number of times 2 appears in the previous 40 entries.
%C A362372 Open question: will all positive integers appear in the sequence?
%H A362372 Michael S. Branicky, <a href="/A362372/b362372.txt">Table of n, a(n) for n = 0..10000</a> (first 4330 terms from Damon Lay)
%e A362372 As an irregular triangle, the table begins:
%e A362372    1;
%e A362372    1;
%e A362372    2, 1;
%e A362372    3, 1, 1;
%e A362372    5, 1, 1;
%e A362372    7, 1, 1;
%e A362372    9, 1, 2;
%e A362372   10, 2, 2;
%e A362372   10, 4, 2, 1, 1;
%e A362372   12, 6, 2, 1, 1, 1, 1;
%e A362372   16, 8, 2, 2, 1, 1, 1, 1, 1, 2;
%e A362372   ...
%e A362372 Initialize the sequence with '1'.
%e A362372 Powers of 1 are counted in the first column, powers of 2 in the second, powers of 3 in the third, etc.
%o A362372 (Python)
%o A362372 from collections import Counter
%o A362372 from sympy import divisors, perfect_power
%o A362372 def powers_in(n):
%o A362372     t = perfect_power(n) # False for n == 1
%o A362372     return [n] if not t else [t[0]**d for d in divisors(t[1])]
%o A362372 def aupton(nn):
%o A362372     num, alst, inventory = 1, [1], Counter([1])
%o A362372     while len(alst) <= nn:
%o A362372         c = inventory[num]
%o A362372         if c == 0: num = 1
%o A362372         else: num += 1; alst.append(c); inventory.update(powers_in(c))
%o A362372     return alst
%o A362372 print(aupton(100)) # _Michael S. Branicky_, May 05 2023
%Y A362372 Cf. A342585 and similar variants thereof: A345730, A347791, A348218, A352799, A353092.
%K A362372 easy,nonn,tabf
%O A362372 0,3
%A A362372 _Damon Lay_, Apr 17 2023