A277790 Numerator of sum of reciprocals of proper divisors of n.

0, 1, 1, 3, 1, 11, 1, 7, 4, 17, 1, 9, 1, 23, 23, 15, 1, 19, 1, 41, 31, 35, 1, 59, 6, 41, 13, 55, 1, 71, 1, 31, 47, 53, 47, 5, 1, 59, 55, 89, 1, 95, 1, 83, 77, 71, 1, 41, 8, 46, 71, 97, 1, 119, 71, 17, 79, 89, 1, 167, 1, 95, 103, 63, 83, 13, 1, 125, 95, 143, 1, 97, 1, 113, 41, 139, 95, 167, 1, 37

Offset

1,4

Links

Antti Karttunen, Table of n, a(n) for n = 1..16384

Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537

Eric Weisstein's World of Mathematics, Restricted Divisor Function

Index entries for sequences related to sums of divisors

Formula

a(n) = numerator(Sum_{d|n, d<n} 1/d).

a(n) = numerator((sigma_1(n)-1)/n).

a(p) = 1 for p is prime.

a(p^k) = (p^k - 1)/(p - 1) for p is prime.

Dirichlet g.f.: (zeta(s) - 1)*zeta(s+1) (for fraction Sum_{d|n, d<n} 1/d).

Example

a(4) = 3 because 4 has 3 divisors {1,2,4} therefore 2 proper divisors {1,2} and 1/1 + 1/2 = 3/2.
0, 1, 1, 3/2, 1, 11/6, 1, 7/4, 4/3, 17/10, 1, 9/4, 1, 23/14, 23/15, 15/8, 1, 19/9, 1, 41/20, 31/21, 35/22, 1, 59/24, 6/5, 41/26, 13/9, 55/28, ...

Maple

with(numtheory): P:=proc(n) local a, k; a:=divisors(n) minus {n};
numer(add(1/a[k], k=1..nops(a))); end: seq(P(i), i=1..80); # Paolo P. Lava, Oct 17 2018

Mathematica

Table[Numerator[DivisorSigma[-1, n] - 1/n], {n, 1, 80}]
Table[Numerator[(DivisorSigma[1, n] - 1)/n], {n, 1, 80}]

Prog

(PARI) a(n) = numerator((sigma(n)-1)/n); \\ Michel Marcus, Nov 01 2016
(Python)
from math import gcd
from sympy import divisor_sigma
def A277790(n): return (m:=divisor_sigma(n)-1)//gcd(n, m) # Chai Wah Wu, Jul 18 2022

Crossrefs

Cf. A000203, A001065, A017665, A017666, A277791 (denominators).

Sequence in context: A276391 A119632 A201131 * A360121 A339175 A134761

Adjacent sequences: A277787 A277788 A277789 * A277791 A277792 A277793

Keyword

nonn,frac

Author

Ilya Gutkovskiy, Oct 31 2016

Status

approved