A389105 - OEIS

%I #68 May 17 2026 12:05:20

%S 1,4,27,8,35,30,847,56,99,250,209,252,2587,1764,75,48,1037,328239090,

%T 58159

%N a(n) is the smallest value of k such that gcd(A381466(k-1), k) = n.

%C Equivalently, a(n) is the first time n is a divisor at a division step of A381466; i.e., a(n) = the smallest value of k such that k/A381466(k) = n.

%C We conjecture that a(n) > 0 for all n; i.e., every positive integer is a divisor in A381466 at least once; indeed, we conjecture that every positive integer is a divisor infinitely many times. For positive values found so far, a(n) > 1.2^(n-1), and we also conjecture this is always true.

%C In the first 10^12 terms of A381466:

%C   - there are 55 known terms up to a(106), see linked a-file;

%C   - the smallest number that has not yet appeared as a divisor is 20;

%C   - 106 is the largest unique divisor to appear at index 2028511506;

%C   - 73 is the last number to appear as a new divisor at index 160322038411.

%H Sam Chapman, <a href="/A389105/a389105.txt">Table of n, a(n) for all terms < 10^12 (where ? denotes missing, larger values)</a>

%e a(2) = 4 because step 4 is the first time we divide by 2 in A381466.

%e a(3) = 27 because step 27 is the first time we divide by 3 in A381466.

%o (PARI) a(n) = my(k=1, v=4, g); while ((g=gcd(k, v)) != n, k++; if (g==1, v += k-1, v = (k-1)/g)); k; \\ _Michel Marcus_, May 17 2026

%Y Cf. A381466.

%K nonn,more

%O 1,2

%A _Sam Chapman_, Mar 11 2026