A277439 Numerators of a sequence defined by a modified recurrence for the exponential of the von Mangoldt function.

1, 2, 3, 4, 5, 6, 7, 16, 27, 20, 11, 12, 13, 56, 135, 64, 85, 18, 19, 320, 567, 352, 115, 144, 175, 832, 1215, 2240, 29, 30, 217, 2560, 8019, 78336, 70, 5184, 925, 1064, 199017, 1120, 451, 42, 5375, 315392, 5670, 329728, 2585, 1152, 91

Offset

1,2

Comments

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.

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

Links

Table of n, a(n) for n=1..49.

Formula

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:

T(1,1) = 1.

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).

Example

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).

Mathematica

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]

Crossrefs

Cf. A014574, A277440 (denominators).

Sequence in context: A332535 A285724 A193551 * A069188 A085158 A065639

Adjacent sequences: A277436 A277437 A277438 * A277440 A277441 A277442

Keyword

nonn,frac

Author

Mats Granvik, Oct 15 2016

Status

approved