cp's OEIS Frontend

A379730 a(1)=1. For n>1 if a(n-1) is a novel term a(n)=A083720(a(n-1)), else a(n) = k*a(n-1), where k(>1) is the number of times a(n-1) has occurred.

%I A379730 #15 Mar 31 2025 22:58:44
%S A379730 1,1,2,1,3,2,4,1,4,8,1,5,6,1,6,12,1,7,30,1,8,16,1,9,2,6,18,1,10,3,6,
%T A379730 24,1,11,210,1,12,24,48,1,13,2310,1,14,15,2,8,24,72,1,15,30,60,1,16,
%U A379730 32,1,17,30030,1,18,36,1,19,510510,1,20,3,9,18,54,1,21
%N A379730 a(1)=1. For n>1 if a(n-1) is a novel term a(n)=A083720(a(n-1)), else a(n) = k*a(n-1), where k(>1) is the number of times a(n-1) has occurred.
%C A379730 In other words if a(n-1) is a novel term, a(n) is the product of all primes < Gpf(a(n-1)) which do not divide a(n-1), else if a(n-1) seen k times already, up to and including itself then a(n)=k*a(n-1). All positive integers appear eventually in the sequence, and the first occurrences of primes appear in order.
%C A379730 Every squarefree number appears infinitely many times (consequent to both conditions of the definition), whereas numbers m which are not squarefree can appear only from the second condition, and therefore appear finitely many (at most A000005(m)-1) times; see Example.
%H A379730 Michael De Vlieger, <a href="/A379730/b379730.txt">Table of n, a(n) for n = 1..10000</a>
%H A379730 Michael De Vlieger, <a href="/A379730/a379730.png">Log log scatterplot of log_10(a(n))</a>, n = 1..2^20.
%F A379730 If a(n-1) is a novel term and rad(a(n-1)) is primorial then a(n)=1.
%e A379730 a(1)=1 is a primorial term so a(2)=1. 1 has now occurred twice so a(3)=2*1=2.
%e A379730 a(3)=2 is a novel primorial term so a(4)=1, and then a(5)=3. Since 3 is novel a(6)=A083720(3)=2.
%e A379730 4 appears twice = A5(4)-1 (as 2*2=a(7), and 4*1=a(9)). It never appears again since A083720(m)!=4 for any m, so 4 can only arise as consequence of the second condition of the definition. Similarly 8 appears only 3 (A5(8)-1) times: 2*4=a(10), 8*1=a(21), and 4*2 a(47). However 16 appears not 5 (A5(16)-1) times but 4 times because 4*4 is prevented since 4 appears only twice.
%t A379730 nn = 120; c[_] := 0; j = 1;
%t A379730 {j}~Join~Reap[Do[
%t A379730   If[c[j] == 0,
%t A379730     k = Product[Prime[i], {i, PrimePi[#[[-1]] ] } ]/Apply[Times, #] &[
%t A379730       FactorInteger[j][[All, 1]] ]; c[j]++,
%t A379730     k = ++c[j]*j ];
%t A379730   j = Sow[k], {nn}] ][[-1, 1]]
%Y A379730 Cf. A000005, A000040, A002110, A005117, A006530, A083720.
%K A379730 nonn,easy
%O A379730 1,3
%A A379730 _David James Sycamore_, Dec 31 2024