OFFSET
1,2
COMMENTS
Denominator(Sum_{prime p | n} 1/p - 1/n) is a factor of n, since all primes in the sum divide n. So a(n) is an integer.
LINKS
Antti Karttunen, Table of n, a(n) for n = 1..20000
Christian Krause, et al, LODA, an assembly language, a computational model and a tool for mining integer sequences
Wikipedia, Giuga number
FORMULA
a(n) = n/A326690(n).
a(n) = n > 1 iff n is either a prime or a Giuga number A007850.
a(n) = gcd(n, 1+((n-1)*A003415(n))). [Conjectured, after an empirical formula found by LODA miner. This holds at least up to n=2^27] - Antti Karttunen, Mar 15 2021
EXAMPLE
a(18) = 18/denominator(Sum_{prime p | 18} 1/p - 1/18) = 18/denominator(1/2 + 1/3 - 1/18) = 18/denominator(7/9) = 18/9 = 2.
a(30) = 30/denominator(Sum_{prime p | 30} 1/p - 1/30) = 30/denominator(1/2 + 1/3 + 1/5 - 1/30) = 30/denominator(1/1) = 30/1 = 30, and 30 is a Giuga number.
MATHEMATICA
PrimeFactors[n_] := Select[Divisors[n], PrimeQ];
f[n_] := Denominator[Sum[1/p, {p, PrimeFactors[n]}] - 1/n];
Table[n/f[n], {n, 79}]
PROG
CROSSREFS
KEYWORD
nonn
AUTHOR
Jonathan Sondow, Jul 20 2019
STATUS
approved