A277791 Denominator of sum of reciprocals of proper divisors of n.
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
Examples
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, ...
Links
- Robert Israel, Table of n, a(n) for n = 1..10000
- Eric Weisstein's World of Mathematics, Restricted Divisor Function
- Index entries for sequences related to sums of divisors
Crossrefs
Programs
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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}] -
PARI
a(n) = denominator((sigma(n)-1)/n); \\ Michel Marcus, Nov 01 2016
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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
Formula
a(n) = denominator(Sum_{d|n, d
a(n) = denominator((sigma_1(n)-1)/n).
a(p) = 1 when p is prime.
a(p^k) = p^(k-1).
Dirichlet g.f.: (zeta(s) - 1)*zeta(s+1) (for fraction Sum_{d|n, d
A355003 a(n) = gcd(A003415(n), A277791(n)), where A003415 is the arithmetic derivative and A277791 is the denominator of sum of reciprocals of proper divisors of n.
1, 1, 1, 2, 1, 1, 1, 4, 3, 1, 1, 4, 1, 1, 1, 8, 1, 3, 1, 4, 1, 1, 1, 4, 5, 1, 9, 4, 1, 1, 1, 16, 1, 1, 1, 2, 1, 1, 1, 4, 1, 1, 1, 4, 3, 1, 1, 16, 7, 5, 1, 4, 1, 27, 1, 4, 1, 1, 1, 4, 1, 1, 3, 32, 1, 1, 1, 4, 1, 1, 1, 12, 1, 1, 5, 4, 1, 1, 1, 16, 27, 1, 1, 4, 1, 1, 1, 4, 1, 3, 1, 4, 1, 1, 1, 16, 1, 7, 3, 5, 1, 1, 1, 4, 1
Offset: 1
Keywords
Links
- Antti Karttunen, Table of n, a(n) for n = 1..65537
Programs
A355818 Greatest common divisor of n, sigma(n) and A276086(n), where A276086 is primorial base exp-function.
1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 5, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 7, 1, 1, 3, 1, 1, 1, 7, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 3
Offset: 1