A263427 Denominator of (prime(prime(n))+2)/(prime(n)+2).

4, 5, 7, 9, 13, 15, 19, 7, 5, 31, 11, 13, 43, 45, 49, 55, 61, 21, 23, 73, 25, 81, 85, 91, 99, 103, 21, 109, 111, 115, 43, 7, 139, 3, 151, 51, 53, 55, 169, 175, 181, 61, 193, 65, 199, 201, 71, 225, 229, 11, 235, 241, 243, 253, 259, 265, 271, 13, 279, 283, 285, 59, 309, 313, 315, 29, 37, 113, 349, 27

Offset

1,1

Comments

We note that a(n) > 1 for all n = 1..10^7 except for n = 394.

Conjecture: For each k = 2,3,6,7,8,9 all the rational numbers (prime(p)+k)/(p+k) with p prime are pairwise distinct.

We have verified that all the numbers (prime(p)+2)/(p+2) with p prime and p < 10^7 are indeed pairwise distinct. Also, for each k = 3,6,7,8,9 we have verified that all the numbers (prime(p)+k)/(p+k) with p prime and p < 3*10^6 are indeed pairwise distinct.

Links

Zhi-Wei Sun, Table of n, a(n) for n = 1..10000

Zhi-Wei Sun, Conjectures on the prime-counting function, a message to Number Theory Mailing List, Oct. 19, 2015.

Example

a(1) = 4 since (prime(prime(1))+2)/(prime(1)+2) = (prime(2)+2)/(2+2) = (3+2)/(2+2) = 5/4.
a(394) = 1 since (prime(prime(394))+2)/(prime(394)+2) = (prime(2707)+2)/(2707+2) = (24379+2)/(2707+2) = 24381/2709 = 9.

Mathematica

p[n_]:=p[n]=Prime[n]
a[n_]:=a[n]=Denominator[(p[p[n]]+2)/(p[n]+2)]
Table[a[n], {n, 1, 70}]

Prog

(PARI) a(n) = denominator((prime(prime(n))+2)/(prime(n)+2));
vector(100, n, a(n)) \\ Altug Alkan, Oct 18 2015

Crossrefs

Cf. A000040, A001359, A006450, A006512, A263399.

Sequence in context: A117150 A239276 A284132 * A052147 A139690 A283559

Adjacent sequences: A263424 A263425 A263426 * A263428 A263429 A263430

Keyword

nonn

Author

Zhi-Wei Sun, Oct 17 2015

Status

approved