A348016 Record the number of terms with no proper divisors, then the number with one proper divisor, then two, three, etc., until reaching a zero term. After each zero term, repeat the count as before.

0, 1, 0, 3, 1, 0, 5, 2, 0, 6, 3, 0, 7, 5, 0, 8, 6, 0, 9, 6, 1, 4, 0, 11, 7, 2, 4, 0, 12, 9, 4, 4, 0, 13, 10, 6, 6, 0, 14, 10, 6, 10, 0, 15, 10, 6, 14, 0, 16, 10, 6, 17, 1, 1, 0, 19, 12, 6, 18, 1, 3, 0, 21, 13, 6, 20, 1, 4, 0, 23, 15, 7, 21, 1, 4, 0, 25, 16, 9, 22, 2, 4, 0, 26, 17

Offset

0,4

Comments

An inventory sequence counting the proper divisors of existing terms, where zero is taken to have no proper divisors (see A032741). After every occurrence of a zero term the incremental count of terms with 0,1,2,... proper divisors is repeated until another zero term is encountered.

Links

David A. Corneth, Table of n, a(n) for n = 0..9999

Example

a(0) = 0 because at first there are no terms, therefore there are no terms with no proper divisors.
a(1) = 1 because now there is one term (a(0)) which has no proper divisors.
a(2) = 0 since there are no terms with one proper divisor.
a(3) = 3 since there are now three terms having just one proper divisor (0,1,0).
As an irregular triangle the sequence begins:
   0, 1, 0;
   3, 1, 0;
   5, 2, 0;
   6, 3, 0;
   7, 5, 0;
   8, 6, 0;
   9, 6, 4, 1, 0;
  11, 7, 2, 4, 0;
  etc.

Prog

(PARI) first(n) = { t = 0; res = vector(n); l = List([1]); for(i = 2, n, for(i = #l + 1, t+1, listput(l, 0) ); res[i] = l[t + 1]; q = if(l[t + 1] == 0, 0, numdiv(l[t + 1]) - 1); for(i = #l + 1, q + 1, listput(l, 0) ); l[q + 1]++; if(res[i] == 0, t = 0 , t++ ) ); res } \\ David A. Corneth, Sep 25 2021
(Python)
from sympy import divisor_count
from collections import Counter
def f(n): return 0 if n == 0 else divisor_count(n) - 1
def aupton(nn):
    num, alst, inventory = 0, [0], Counter([0])
    for n in range(1, nn+1):
        c = inventory[num]
        num = 0 if c == 0 else num + 1
        alst.append(c)
        inventory.update([f(c)])
    return alst
print(aupton(84)) # Michael S. Branicky, May 07 2023

Crossrefs

Cf. A032741, A342585, A345730, A347791.

Sequence in context: A245095 A154791 A121440 * A353092 A362564 A324664

Adjacent sequences: A348013 A348014 A348015 * A348017 A348018 A348019

Keyword

nonn,tabf

Author

David James Sycamore, Sep 24 2021

Extensions

Data corrected and extended by David A. Corneth, Sep 25 2021

Status

approved