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
KEYWORD
nonn
AUTHOR
Zhi-Wei Sun, Oct 17 2015
STATUS
approved