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A277790
Numerator of sum of reciprocals of proper divisors of n.
4
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
FORMULA
a(n) = numerator(Sum_{d|n, d<n} 1/d).
a(n) = numerator((sigma_1(n)-1)/n).
a(p) = 1 when p is prime.
a(p^k) = (p^k - 1)/(p - 1) when 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
KEYWORD
nonn,frac
AUTHOR
Ilya Gutkovskiy, Oct 31 2016
STATUS
approved