

A277439


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


2



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
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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 A277439(n)/A277440(n) is defined by the recurrence:
T(1,1) = 1.
T(n,k) = if k = 1 then n/(Product_{i=1..n1}(T(n + 1, k + i)))/(Product_{i=1..n1}(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 integers are 1,2,4,6,12,... which for n>2 is equal to A014574.


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: A132028 A285724 A193551 * A069188 A085158 A065639
Adjacent sequences: A277436 A277437 A277438 * A277440 A277441 A277442


KEYWORD

nonn,frac


AUTHOR

Mats Granvik, Oct 15 2016


STATUS

approved



