This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A347791 #24 Nov 08 2021 06:44:23
%S A347791 0,1,1,0,2,2,2,0,3,2,5,0,4,2,7,1,0,5,3,10,2,0,6,3,12,3,1,0,7,4,14,5,1,
%T A347791 0,8,5,16,5,2,1,0,9,6,18,7,3,1,0,10,7,21,9,3,1,0,11,8,23,10,4,1,0,12,
%U A347791 9,24,13,5,2,0,13,9,28,14,6,2,0,14,9,29,18,7
%N A347791 Inventory sequence using prime divisors (with multiplicity): Record the number of terms thus far which are divisible by every prime, then the number of terms thus far not divisible by any prime, then the number divisible (once) by a single prime, then 2 (including with multiplicity), then 3 etc until a zero is recorded. Repeat after every zero term.
%C A347791 Inspired by the original Inventory sequence (A342585). It follows from the definition that zero appears infinitely many times. Every nonzero number number appears because the number of zero terms increments at every extension of the sequence between consecutive zeros. Since the count for each number of divisors increments (eventually) as the sequence extends, it follows that every number appears infinitely many times.
%C A347791 The count of terms divisible by every prime is the number of zeros, the count of terms divisible by no prime is the number of 1s, the count of terms divisible by one prime (once) is the number of primes, etc.
%H A347791 Michael De Vlieger, <a href="/A347791/b347791.txt">Table of n, a(n) for n = 0..10000</a> (rows 0 <= n <= 836, flattened)
%H A347791 Michael De Vlieger, <a href="/A347791/a347791.png">Log-log scatterplot of row n</a> for 1 <= n <= 2^16 (1199674 terms), with color function showing k, black represents the number of 0s, red 1s, orange primes, etc.
%e A347791 a(0)=0 because at this point there are zero terms divisible by every prime. a(1)=1 because there is now one term (0) which is divisible by every prime. a(2)=1 because there is now one term (1) with no prime divisor. a(3)=0 because at this point there is no term divisible by one prime, and so on.
%e A347791 As an irregular triangle the sequence begins:
%e A347791 0;
%e A347791 1, 1, 0;
%e A347791 2, 2, 2, 0;
%e A347791 3, 2, 5, 0;
%e A347791 4, 2, 7, 1, 0;
%e A347791 5, 3, 10, 2, 0;
%e A347791 6, 3, 12, 3, 1, 0;
%e A347791 7, 4, 14, 5, 1, 0;
%e A347791 8, 5, 16, 5, 2, 1, 0;
%e A347791 9, 6, 18, 7, 3, 1, 0;
%e A347791 10, 7, 21, 9, 3, 1, 0;
%e A347791 11, 8, 23, 10, 4, 1, 0; etc.
%t A347791 Block[{a = {}, c, k, m}, c[-1] = 0; Do[k = -1; c[-1]++; AppendTo[a, 0]; While[IntegerQ[c[k]], AppendTo[a, c[k]]; Set[m, PrimeOmega[c[k]]]; If[IntegerQ[c[m]], c[m]++, Set[c[m], 1]]; k++], 10]; a] (* _Michael De Vlieger_, Sep 15 2021 *)
%Y A347791 Cf. A001222, A342585, A347738.
%K A347791 nonn,tabf
%O A347791 0,5
%A A347791 _David James Sycamore_, Sep 13 2021