login
A277791
Denominator of sum of reciprocals of proper divisors of n.
7
1, 1, 1, 2, 1, 6, 1, 4, 3, 10, 1, 4, 1, 14, 15, 8, 1, 9, 1, 20, 21, 22, 1, 24, 5, 26, 9, 28, 1, 30, 1, 16, 33, 34, 35, 2, 1, 38, 39, 40, 1, 42, 1, 44, 45, 46, 1, 16, 7, 25, 51, 52, 1, 54, 55, 8, 57, 58, 1, 60, 1, 62, 63, 32, 65, 6, 1, 68, 69, 70, 1, 36, 1, 74, 25, 76, 77, 78, 1, 16
OFFSET
1,4
FORMULA
a(n) = denominator(Sum_{d|n, d<n} 1/d).
a(n) = denominator((sigma_1(n)-1)/n).
a(p) = 1 for p is prime.
a(p^k) = p^(k-1).
Dirichlet g.f.: (zeta(s) - 1)*zeta(s+1) (for fraction Sum_{d|n, d<n} 1/d).
EXAMPLE
a(4) = 2 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};
denom(add(1/a[k], k=1..nops(a))); end: seq(P(i), i=1..80); # Paolo P. Lava, Oct 17 2018
MATHEMATICA
Table[Denominator[DivisorSigma[-1, n] - 1/n], {n, 1, 80}]
Table[Denominator[(DivisorSigma[1, n] - 1)/n], {n, 1, 80}]
PROG
(PARI) a(n) = denominator((sigma(n)-1)/n); \\ Michel Marcus, Nov 01 2016
(Python)
from math import gcd
from sympy import divisor_sigma
def A277791(n): return n//gcd(n, divisor_sigma(n)-1) # Chai Wah Wu, Jul 18 2022
CROSSREFS
Cf. A000203, A001065, A017665, A017666, A277790 (numerators), A281086, A355003, A355694 (Dirichlet inverse), A355815.
Sequence in context: A242926 A189733 A306927 * A243146 A349440 A048671
KEYWORD
nonn,frac,look
AUTHOR
Ilya Gutkovskiy, Oct 31 2016
STATUS
approved