A378722 - OEIS

%I #13 Dec 07 2024 08:02:31

%S 1,2,3,4,6,9,12,18,15,5,7,8,10,20,30,45,60,90,75,25,35,14,21,28,42,63,

%T 84,126,105,70,140,210,315,420,630,525,175,245,49,56,98,77,11,13,16,

%U 22,33,44,66,99,132,198,165,55,88,110,220,330,495,660,990,825,275

%N a(1) = 1, a(2) = 2. For n > 2, a(n) is the smallest number not already a term which is divisible by A007947(a(n-1)/A020639(a(n-1))).

%C In other words a(n) is the least novel multiple of the squarefree kernel of a(n-1)/p, where p is the smallest prime factor of a(n-1).

%C If a(n-1) is prime, a(n) is the smallest unused number in the sequence so far.

%C Conjectured to be a permutation of the positive integers with primes in order.

%H Michael De Vlieger, <a href="/A378722/b378722.txt">Table of n, a(n) for n = 1..10000</a>

%H Michael De Vlieger, <a href="/A378722/a378722.png">Log log scatterplot of a(n)</a>, n = 1..2^20.

%H Michael De Vlieger, <a href="/A378722/a378722_1.png">Log log scatterplot of a(n)</a>, n = 1..2^14, showing primes in red, proper prime powers in gold, squarefree composites in green, and numbers neither squarefree nor prime powers in blue and purple, where purple also shows powerful numbers that are not prime powers.

%e a(2)= 2 (given), therefore a(3) is the smallest number not already a term which is divisible by the squarefree kernel of 2/2 = 1. Thus a(3) = 3. Likewise a(4) = 4. Since A007947(4/2) = 2, a(5) = 6, the least novel multiple of 2.

%t nn = 120; c[_] := False; m[_] := 1;

%t Do[Set[{a[i], c[i]}, {i, True}], {i, 2}]; j = a[2];

%t Do[k = Times @@ FactorInteger[j/FactorInteger[j][[1, 1]] ][[All, 1]];

%t   While[c[k*m[k]], m[k]++]; k *= m[k];

%t   Set[{a[n], c[k], j}, {k, True, k}], {n, 3, nn}];

%t Array[a, nn] (* _Michael De Vlieger_, Dec 06 2024 *)

%Y Cf. A007947, A020639.

%K nonn,easy,new

%O 1,2

%A _David James Sycamore_, Dec 05 2024

%E More terms from _Michael De Vlieger_, Dec 06 2024.