cp's OEIS Frontend

A354827 Numerators of Dirichlet inverse of fraction A003961(n) / sigma(n).

%I A354827 #11 Jun 07 2022 16:23:22
%S A354827 1,-1,-5,-2,-7,5,-11,-8,-75,7,-13,5,-17,11,35,-1648,-19,75,-23,1,55,
%T A354827 13,-29,2,-245,17,-225,11,-31,-35,-37,-1664,65,19,77,75,-41,23,85,4,
%U A354827 -43,-55,-47,13,175,29,-53,412,-847,245,95,17,-59,225,91,11,23,31,-61,-5,-67,37,825,-7662464,17,-65,-71,19,145,-77
%N A354827 Numerators of Dirichlet inverse of fraction A003961(n) / sigma(n).
%C A354827 Because the ratio A003961(n) / A000203(n) is multiplicative, so is also its Dirichlet inverse (which also is a sequence of rational numbers). This sequence gives the numerators when presented in its lowest terms, while A354828 gives the denominators. See the examples.
%H A354827 Antti Karttunen, <a href="/A354827/b354827.txt">Table of n, a(n) for n = 1..65537</a>
%H A354827 <a href="/index/Pri#prime_indices">Index entries for sequences computed from indices in prime factorization</a>
%H A354827 <a href="/index/Si#SIGMAN">Index entries for sequences related to sigma(n)</a>
%e A354827 The ratio a(n)/A354828(n) for n = 1..21: 1, -1, -5/4, -2/7, -7/6, 5/4, -11/8, -8/35, -75/208, 7/6, -13/12, 5/14, -17/14, 11/8, 35/24, -1648/7595, -19/18, 75/208, -23/20, 1/3, 55/32.
%o A354827 (PARI)
%o A354827 up_to = 65537;
%o A354827 A003961(n) = my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); \\ From A003961
%o A354827 DirInverseCorrect(v) = { my(u=vector(#v)); u[1] = (1/v[1]); for(n=2, #v, u[n] = (-u[1]*sumdiv(n, d, if(d<n, v[n/d]*u[d], 0)))); (u) }; \\ Compute the Dirichlet inverse of the sequence given in input vector v.
%o A354827 AuxA354827(n) = (A003961(n)/sigma(n));
%o A354827 vDirInv = DirInverseCorrect(vector(up_to,n,AuxA354827(n)));
%o A354827 A354827(n) = numerator(vDirInv[n]);
%o A354827 A354828(n) = denominator(vDirInv[n]);
%Y A354827 Cf. A000203, A003961, A349161, A349162.
%Y A354827 Cf. A354828 (denominators).
%Y A354827 Cf. also A349627, A354365.
%K A354827 sign,frac
%O A354827 1,3
%A A354827 _Antti Karttunen_, Jun 07 2022