A265710 a(n) = denominator of Sum_{d|n} 1/sigma(d).

1, 3, 4, 21, 6, 3, 8, 35, 52, 9, 12, 84, 14, 2, 24, 1085, 18, 13, 20, 18, 32, 9, 24, 14, 186, 7, 520, 56, 30, 18, 32, 9765, 48, 27, 16, 364, 38, 5, 56, 5, 42, 8, 44, 252, 104, 18, 48, 868, 456, 279, 72, 98, 54, 390, 72, 140, 16, 45, 60, 72, 62, 8, 416, 1240155

Offset: 1

Author

Jaroslav Krizek, Dec 24 2015

Keywords

• nonn
• frac

Comments

a(n) = denominator of Sum_{d|n} 1/A000203(d).

Are there numbers n > 1 such that Sum_{d|n} 1/sigma(d) is an integer?

a(n) = 2 for n = 14, 244, 494, 45994. Are there any others? - Robert Israel, Apr 02 2017

Examples

			For n = 6; divisors d of 6: {1, 2, 3, 6}; sigma(d): {1, 3, 4, 12}; Sum_{d|6} 1/sigma(d) = 1/1 + 1/3 + 1/4 + 1/12 = 20/12 = 5/3; a(n) = 3.
		

Links

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

Crossrefs

Cf. A069934, A000203, A265708, A265709, A265711, A265712, A265713, A265714, A266227, A266228.

Programs

Maple

f:= n -> denom(add(1/numtheory:-sigma(d), d = numtheory:-divisors(n))):
map(f, [$1..200]); # Robert Israel, Apr 02 2017

Mathematica

Table[Denominator[Plus@@(1/DivisorSigma[1, Divisors[n]])], {n, 70}] (* Alonso del Arte, Dec 24 2015 *)

PARI

a(n) = denominator(sumdiv(n, d, 1/sigma(d))); \\ Michel Marcus, Feb 06 2024

Formula

a(1) = 1; a(p) = p + 1 for p = prime.

a(n) = A265709(n) / (Sum_{d|n} 1/sigma(d)) = A265709(n) * A069934(n) / A265708(n).