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A277790 Numerator of sum of reciprocals of proper divisors of n.

%I A277790 #23 Feb 16 2025 08:33:37
%S A277790 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,
%T A277790 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,
%U A277790 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
%N A277790 Numerator of sum of reciprocals of proper divisors of n.
%H A277790 Antti Karttunen, <a href="/A277790/b277790.txt">Table of n, a(n) for n = 1..16384</a>
%H A277790 Antti Karttunen, <a href="/A277790/a277790.txt">Data supplement: n, a(n) computed for n = 1..65537</a>
%H A277790 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RestrictedDivisorFunction.html">Restricted Divisor Function</a>
%H A277790 <a href="/index/Su#sums_of_divisors">Index entries for sequences related to sums of divisors</a>
%F A277790 a(n) = numerator(Sum_{d|n, d<n} 1/d).
%F A277790 a(n) = numerator((sigma_1(n)-1)/n).
%F A277790 a(p) = 1 when p is prime.
%F A277790 a(p^k) = (p^k - 1)/(p - 1) when p is prime.
%F A277790 Dirichlet g.f.: (zeta(s) - 1)*zeta(s+1) (for fraction Sum_{d|n, d<n} 1/d).
%e A277790 a(4) = 3 because 4 has 3 divisors {1,2,4} therefore 2 proper divisors {1,2} and 1/1 + 1/2 = 3/2.
%e A277790 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 A277790 with(numtheory): P:=proc(n) local a,k; a:=divisors(n) minus {n};
%p A277790 numer(add(1/a[k],k=1..nops(a))); end: seq(P(i),i=1..80); # _Paolo P. Lava_, Oct 17 2018
%t A277790 Table[Numerator[DivisorSigma[-1, n] - 1/n], {n, 1, 80}]
%t A277790 Table[Numerator[(DivisorSigma[1, n] - 1)/n], {n, 1, 80}]
%o A277790 (PARI) a(n) = numerator((sigma(n)-1)/n); \\ _Michel Marcus_, Nov 01 2016
%o A277790 (Python)
%o A277790 from math import gcd
%o A277790 from sympy import divisor_sigma
%o A277790 def A277790(n): return (m:=divisor_sigma(n)-1)//gcd(n,m) # _Chai Wah Wu_, Jul 18 2022
%Y A277790 Cf. A000203, A001065, A017665, A017666, A277791 (denominators).
%K A277790 nonn,frac
%O A277790 1,4
%A A277790 _Ilya Gutkovskiy_, Oct 31 2016