A354365 - OEIS

%I #16 Jun 08 2022 10:18:32

%S 1,-2,-3,0,-5,3,-7,0,0,10,-11,0,-13,14,5,0,-17,0,-19,0,21,22,-23,0,0,

%T 26,0,0,-29,-5,-31,0,33,34,7,0,-37,38,39,0,-41,-21,-43,0,0,46,-47,0,0,

%U 0,51,0,-53,0,55,0,57,58,-59,0,-61,62,0,0,65,-33,-67,0,69,-14,-71,0,-73,74,0,0,11,-39,-79,0,0,82

%N Numerators of Dirichlet inverse of primorial deflation fraction A319626(n) / A319627(n).

%C Because the ratio n / A064989(n) = A319626(n) / A319627(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 A354366 gives the denominators. See the examples.

%H <a href="/index/Pri#prime_indices">Index entries for sequences computed from indices in prime factorization</a>

%F a(n) = A055615(n) / gcd(A055615(n), A064989(n)).

%e The ratio a(n)/A354366(n) for n = 1..22: 1, -2, -3/2, 0, -5/3, 3, -7/5, 0, 0, 10/3, -11/7, 0, -13/11, 14/5, 5/2, 0, -17/13, 0, -19/17, 0, 21/10, 22/7.

%o (PARI)

%o A064989(n) = { my(f = factor(n)); if((n>1 && f[1,1]==2), f[1,2] = 0); for (i=1, #f~, f[i,1] = precprime(f[i,1]-1)); factorback(f); };

%o A354365(n) = numerator((moebius(n)*n)/A064989(n));

%Y Cf. A013929 (positions of 0's), A055615, A319626, A319627, A354350.

%Y Cf. A354366 (denominators).

%Y Cf. also A349629, A354351, A354827.

%K sign,frac

%O 1,2

%A _Antti Karttunen_, Jun 07 2022