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 A384533 #13 Jun 22 2025 00:51:20
%S A384533 1,1,0,2,1,0,3,2,0,3,4,1,0,4,4,3,0,4,5,4,0,4,6,7,0,4,7,8,1,0,5,9,9,1,
%T A384533 0,6,9,12,2,0,6,10,14,2,0,6,11,16,2,1,0,7,14,17,2,1,0,8,16,17,3,2,0,8,
%U A384533 19,17,4,2,0,8,22,19,5,2,0,8,25,20,7,2,0,8
%N A384533 Inventory sequence starting a(1) = 1 and thereafter recording the number of terms which are the product of 0,1,2,... primes (taken with multiplicity). The count resets following a term = 0.
%C A384533 When for some k, a(k) = 0, a(k+1) is the number 1's (empty products) in the sequence. The number of zero terms are never counted because there are no numbers whose product of prime factors = 0.
%H A384533 Michael De Vlieger, <a href="/A384533/b384533.txt">Table of n, a(n) for n = 1..10000</a>
%H A384533 Michael De Vlieger, <a href="/A384533/a384533.png">Log log scatterplot of a(n)</a>, n = 1..2^20, replacing a(n) = 0 with 1/2 for visibility.
%H A384533 Michael De Vlieger, <a href="/A384533/a384533_1.png">Log log scatterplot of a(n)</a>, n = 1..2^16, replacing a(n) = 0 with 1/2 for visibility, with a color function showing counter c = 0 in black, c = 1 in red, c = 2 in orange, .. c = 14 in magenta.
%e A384533 a(1) = 1, so a(2) = 1 since there is one term which is the product of no primes.
%e A384533 a(3) = 0 since there are no terms which are the product of one prime.
%e A384533 Now the count resets and a(4) = 2 since there are two empty products (a(1) = a(2) = 1).
%e A384533 a(5) = 1 since a(4) = 2 is the product of one prime.
%e A384533 a(6) = 0 since there are no terms which are the product of two primes.
%e A384533 The count resets again and a(7) = 3, the number of empty products.
%e A384533 a(8) = 2 because there are now two terms (a(4) = 2 and a(7) = 3) which are the product of one prime.
%e A384533 Since there are still no terms which are the product of two primes, a(9) = 0.
%e A384533 a(10) = 3, still the number of empty products and a(11) = 4 since there are now four prime terms. This is the first term with two prime divisors (counted with multiplicity), implying that a(12), the next count of two prime divisors must be = 1. And so on.
%t A384533 nn = 120; q[_] := 0; j = 1; c = 0; q[0]++; {j}~Join~Reap[Do[If[j == 0, c = 0]; j = Sow[q[c]]; c++; If[j > 0, q[PrimeOmega[j]]++], {n, 2, nn}] ][[-1, 1]] (* _Michael De Vlieger_, Jun 02 2025 *)
%Y A384533 Cf. A342585.
%K A384533 nonn,easy
%O A384533 1,4
%A A384533 _David James Sycamore_, Jun 01 2025