A326691 a(n) = n/denominator(Sum_{prime p | n} 1/p - 1/n).

1, 2, 3, 1, 5, 2, 7, 1, 1, 2, 11, 3, 13, 2, 1, 1, 17, 2, 19, 1, 3, 2, 23, 1, 1, 2, 1, 1, 29, 30, 31, 1, 1, 2, 1, 1, 37, 2, 3, 1, 41, 2, 43, 1, 1, 2, 47, 3, 1, 2, 1, 1, 53, 2, 5, 7, 3, 2, 59, 1, 61, 2, 1, 1, 1, 6, 67, 1, 1, 2, 71, 1, 73, 2, 3, 1, 1, 2, 79

Offset: 1

Author

Jonathan Sondow, Jul 20 2019

Keywords

• nonn

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.

Examples

			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.
		

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

Crossrefs

Cf. A003415, A007850, A326689, A326690, A326692, A326715.

Programs

Mathematica

PrimeFactors[n_] := Select[Divisors[n], PrimeQ];
f[n_] := Denominator[Sum[1/p, {p, PrimeFactors[n]}] - 1/n];
Table[n/f[n], {n, 79}]

PARI

A326691(n) = (n/A326690(n)); \\ Antti Karttunen, Mar 15 2021

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