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A277791 Denominator of sum of reciprocals of proper divisors of n.

%I A277791 #25 Feb 16 2025 08:33:37
%S A277791 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,
%T A277791 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,
%U A277791 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
%N A277791 Denominator of sum of reciprocals of proper divisors of n.
%H A277791 Robert Israel, <a href="/A277791/b277791.txt">Table of n, a(n) for n = 1..10000</a>
%H A277791 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RestrictedDivisorFunction.html">Restricted Divisor Function</a>
%H A277791 <a href="/index/Su#sums_of_divisors">Index entries for sequences related to sums of divisors</a>
%F A277791 a(n) = denominator(Sum_{d|n, d<n} 1/d).
%F A277791 a(n) = denominator((sigma_1(n)-1)/n).
%F A277791 a(p) = 1 when p is prime.
%F A277791 a(p^k) = p^(k-1).
%F A277791 Dirichlet g.f.: (zeta(s) - 1)*zeta(s+1) (for fraction Sum_{d|n, d<n} 1/d).
%e A277791 a(4) = 2 because 4 has 3 divisors {1,2,4} therefore 2 proper divisors {1,2} and 1/1 + 1/2 = 3/2.
%e A277791 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, ...
%p A277791 with(numtheory): P:=proc(n) local a,k; a:=divisors(n) minus {n};
%p A277791 denom(add(1/a[k],k=1..nops(a))); end: seq(P(i),i=1..80); # _Paolo P. Lava_, Oct 17 2018
%t A277791 Table[Denominator[DivisorSigma[-1, n] - 1/n], {n, 1, 80}]
%t A277791 Table[Denominator[(DivisorSigma[1, n] - 1)/n], {n, 1, 80}]
%o A277791 (PARI) a(n) = denominator((sigma(n)-1)/n); \\ _Michel Marcus_, Nov 01 2016
%o A277791 (Python)
%o A277791 from math import gcd
%o A277791 from sympy import divisor_sigma
%o A277791 def A277791(n): return n//gcd(n,divisor_sigma(n)-1) # _Chai Wah Wu_, Jul 18 2022
%Y A277791 Cf. A000203, A001065, A017665, A017666, A277790 (numerators), A281086, A355003, A355694 (Dirichlet inverse), A355815.
%K A277791 nonn,frac,look
%O A277791 1,4
%A A277791 _Ilya Gutkovskiy_, Oct 31 2016