A277439 - OEIS

%I #17 Dec 16 2019 10:24:04

%S 1,2,3,4,5,6,7,16,27,20,11,12,13,56,135,64,85,18,19,320,567,352,115,

%T 144,175,832,1215,2240,29,30,217,2560,8019,78336,70,5184,925,1064,

%U 199017,1120,451,42,5375,315392,5670,329728,2585,1152,91

%N Numerators of a sequence defined by a modified recurrence for the exponential of the von Mangoldt function.

%C If n is equal to the average of twin prime pairs then the ratio A277439(n)/A277440(n) is equal to n by definition of the recurrence.

%C Conjecture for n > 2: The ratio is equal to n if and only if n is the average of twin prime pairs.

%F Numerators of the ratio a(n)/A277440(n), which is the first column of the array (T(n,k): n,k >= 1) that is defined by the following recurrence:

%F T(1,1) = 1.

%F T(n,k) = if k = 1 then n/(Product_{i=1..n-1}(T(n + 1, k + i)))/(Product_{i=1..n-1}(T(n - 1, k + i))) else if(mod(n, k) = 0 then T(n/k, 1) else 1) else 1).

%e The ratio starts: 1, 2, 3/2, 4, 5/6, 6, 7/24, 16/3, 27/40, 20/3, 11/120, 12, 13/42, ..., where the integer terms are 1, 2, 4, 6, 12,.... For n > 2, the latter sequence equals A014574 (average of twin primes).

%t Clear[t]; nn = 49; t[1, 1] = 1; t[n_, k_] := t[n, k] = If[k == 1, n/Product[t[n + 1, k + i], {i, 1, n - 1}]/Product[t[n - 1, k + i], {i, 1, n - 1}], If[Mod[n, k] == 0, t[n/k, 1], 1], 1]; a = Table[t[n, 1], {n, 1, nn}]; Denominator[a]; Numerator[a]

%Y Cf. A014574, A277440 (denominators).

%K nonn,frac

%O 1,2

%A _Mats Granvik_, Oct 15 2016