A326715 Values of n for which the denominator of (Sum_{prime p | n} 1/p - 1/n) is 1.

1, 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 30, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241

Offset: 1

Author

Jonathan Sondow, Jul 20 2019

Keywords

• nonn

Comments

n is in the sequence iff either n = 1 or n is a prime or n is a Giuga number, by one definition of Giuga numbers A007850.

Examples

			a(30) = denominator(Sum_{prime p | 30} 1/p - 1/30) = denominator(1/2 + 1/3 + 1/5 - 1/30) = denominator(1/1) = 1, and 30 is a Giuga number.
		

Links

• Robert Israel, Table of n, a(n) for n = 1..10000
• Wikipedia, Giuga number

Crossrefs

Cf. A007850, A326689, A326690, A326691, A326692.

Programs

Maple

filter:= proc(n) local p;
   denom(add(1/p, p = numtheory:-factorset(n))-1/n)=1
end proc:
select(filter, [$1..300]); # Robert Israel, Dec 15 2020

Mathematica

PrimeFactors[n_] := Select[Divisors[n], PrimeQ];
f[n_] := Denominator[Sum[1/p, {p, PrimeFactors[n]}] - 1/n];
Select[Range[148], f[#] == 1 &]

Formula

n such that A326690(n) = 1.