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
A355815 a(n) = gcd(A276086(n), A277791(n)), where A276086 is primorial base exp-function and A277791 is the denominator of sum of reciprocals of proper divisors of n.
1, 1, 1, 1, 1, 1, 1, 1, 3, 5, 1, 1, 1, 1, 15, 1, 1, 1, 1, 5, 3, 1, 1, 1, 5, 1, 3, 1, 1, 1, 1, 1, 3, 1, 7, 1, 1, 1, 3, 5, 1, 7, 1, 1, 15, 1, 1, 1, 7, 25, 3, 1, 1, 1, 5, 1, 3, 1, 1, 1, 1, 1, 21, 1, 1, 1, 1, 1, 3, 35, 1, 1, 1, 1, 25, 1, 7, 1, 1, 1, 3, 1, 1, 7, 5, 1, 3, 1, 1, 1, 7, 1, 3, 1, 1, 1, 1, 49, 3, 5, 1, 1, 1, 1, 105
Offset: 1
Keywords
Links
- Antti Karttunen, Table of n, a(n) for n = 1..11550
- Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537
- Index entries for sequences related to primorial base
Programs
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PARI
A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); }; A277791(n) = denominator((sigma(n)-1)/n); \\ From A277791 A355815(n) = gcd(A276086(n), A277791(n));
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Python
from math import gcd from sympy import nextprime, divisor_sigma def A355815(n): m, p, c = 1, 2, n while c: c, a = divmod(c,p) m *= p**a p = nextprime(p) return gcd(m,n//gcd(n, divisor_sigma(n)-1)) # Chai Wah Wu, Jul 18 2022