A349397 - OEIS

%I #12 Nov 25 2021 20:30:06

%S 1,0,0,0,0,-1,5,-8,0,6,3,-2,0,-19,5,4,4,-20,19,-22,-6,15,-3,8,0,0,-16,

%T -16,18,-24,40,-70,-9,24,-21,8,50,-55,-8,24,-6,31,15,-58,-20,17,31,

%U -92,-2,-70,37,24,0,20,49,18,-6,-26,13,-33,15,-62,-158,-20,22,-15,49,-130,67,48,49,-58,29,-112,-4,60,-73,-16

%N Dirichlet convolution of A064216 with the Dirichlet inverse of its inverse permutation.

%C Dirichlet convolution of A064216 with A323893, which is the Dirichlet inverse of A048673. Therefore, convolving A048673 with this sequence gives A064216.

%C Note how for n = 1 .. 35, a(n) = -A349398(n).

%H Antti Karttunen, <a href="/A349397/b349397.txt">Table of n, a(n) for n = 1..16384</a>

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

%F a(n) = Sum_{d|n} A064216(n/d) * A323893(d).

%o (PARI)

%o A048673(n) = { my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); (1/2)*(1+factorback(f)); };

%o A064216(n) = { my(f = factor(n+n-1)); 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 memoA323893 = Map();

%o A323893(n) = if(1==n,1,my(v); if(mapisdefined(memoA323893,n,&v), v, v = -sumdiv(n,d,if(d<n,A048673(n/d)*A323893(d),0)); mapput(memoA323893,n,v); (v)));

%o A349397(n) = sumdiv(n,d,A064216(n/d)*A323893(d));

%Y Cf. A003961, A048673, A064216, A064989, A323893, A349398 (Dirichlet inverse), A349399 (sum with it), A349384.

%Y Cf. also pairs A349376, A349377 and A349613, A349614 for similar constructions.

%K sign

%O 1,7

%A _Antti Karttunen_, Nov 19 2021