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.

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, 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, 2, 2, 1, 1, 1, 1, 1, 2, 36, 21, 2, 2, 1, 2, 1, 1, 1

Offset

0,3

Comments

A variant of the inventory sequence, A342585.

The graph exhibits sharp jumps followed by a rapid decline forming a periodic hockey stick pattern. Larger-scale, near-linear structures also appear.

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.

Open question: will all positive integers appear in the sequence?

Links

Michael S. Branicky, Table of n, a(n) for n = 0..10000 (first 4330 terms from Damon Lay)

Example

As an irregular triangle, the table begins:
   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, 8, 2, 2, 1, 1, 1, 1, 1, 2;
  ...
Initialize the sequence with '1'.
Powers of 1 are counted in the first column, powers of 2 in the second, powers of 3 in the third, etc.

Prog

(Python)
from collections import Counter
from sympy import divisors, perfect_power
def powers_in(n):
    t = perfect_power(n) # False for n == 1
    return [n] if not t else [t[0]**d for d in divisors(t[1])]
def aupton(nn):
    num, alst, inventory = 1, [1], Counter([1])
    while len(alst) <= nn:
        c = inventory[num]
        if c == 0: num = 1
        else: num += 1; alst.append(c); inventory.update(powers_in(c))
    return alst
print(aupton(100)) # Michael S. Branicky, May 05 2023

Crossrefs

Cf. A342585 and similar variants thereof: A345730, A347791, A348218, A352799, A353092.

Sequence in context: A141412 A178623 A210765 * A160183 A168534 A026807

Adjacent sequences: A362369 A362370 A362371 * A362373 A362374 A362375

Keyword

easy,nonn,tabf

Author

Damon Lay, Apr 17 2023

Status

approved