cp's OEIS Frontend

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.

%I A348016 #20 May 07 2023 09:59:07
%S A348016 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,
%T A348016 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,
%U A348016 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
%N 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.
%C A348016 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.
%H A348016 David A. Corneth, <a href="/A348016/b348016.txt">Table of n, a(n) for n = 0..9999</a>
%e A348016 a(0) = 0 because at first there are no terms, therefore there are no terms with no proper divisors.
%e A348016 a(1) = 1 because now there is one term (a(0)) which has no proper divisors.
%e A348016 a(2) = 0 since there are no terms with one proper divisor.
%e A348016 a(3) = 3 since there are now three terms having just one proper divisor (0,1,0).
%e A348016 As an irregular triangle the sequence begins:
%e A348016    0, 1, 0;
%e A348016    3, 1, 0;
%e A348016    5, 2, 0;
%e A348016    6, 3, 0;
%e A348016    7, 5, 0;
%e A348016    8, 6, 0;
%e A348016    9, 6, 4, 1, 0;
%e A348016   11, 7, 2, 4, 0;
%e A348016   etc.
%o A348016 (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
%o A348016 (Python)
%o A348016 from sympy import divisor_count
%o A348016 from collections import Counter
%o A348016 def f(n): return 0 if n == 0 else divisor_count(n) - 1
%o A348016 def aupton(nn):
%o A348016     num, alst, inventory = 0, [0], Counter([0])
%o A348016     for n in range(1, nn+1):
%o A348016         c = inventory[num]
%o A348016         num = 0 if c == 0 else num + 1
%o A348016         alst.append(c)
%o A348016         inventory.update([f(c)])
%o A348016     return alst
%o A348016 print(aupton(84)) # _Michael S. Branicky_, May 07 2023
%Y A348016 Cf. A032741, A342585, A345730, A347791.
%K A348016 nonn,tabf
%O A348016 0,4
%A A348016 _David James Sycamore_, Sep 24 2021
%E A348016 Data corrected and extended by _David A. Corneth_, Sep 25 2021