A349398 Dirichlet convolution of A048673 with the Dirichlet inverse of its inverse permutation.

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, 16, -18, 24, -40, 70, 9, -24, 21, -7, -50, 55, 8, -24, 6, -41, -15, 58, 20, -17, -31, 108, 27, 70, -37, -24, 0, -20, -49, -98, 6, 26, -13, 21, -15, 62, 158, 84, -22, 9, -49, 130, -67, 12, -49, 62, -29, 112, 4, -60, 103, 16

Offset: 1

Author

Antti Karttunen, Nov 19 2021

Keywords

• sign

Comments

Dirichlet convolution of A048673 with A349358, which is the Dirichlet inverse of A064216 (inverse permutation of A048673). Therefore, convolving A064216 with this sequence gives A048673.

Note how for n = 1 .. 35, a(n) = -A349397(n).

Links

• Antti Karttunen, Table of n, a(n) for n = 1..16384
• Index entries for sequences computed from indices in prime factorization

Crossrefs

Cf. A003961, A048673, A064216, A064989, A323893, A349397 (Dirichlet inverse), A349399 (sum with it).

Cf. also A349376, A349377, A349385.

Programs

PARI

A048673(n) = { my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); (1/2)*(1+factorback(f)); };
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); };
memoA349358 = Map();
A349358(n) = if(1==n,1,my(v); if(mapisdefined(memoA349358,n,&v), v, v = -sumdiv(n,d,if(dA064216(n/d)*A349358(d),0)); mapput(memoA349358,n,v); (v)));
A349398(n) = sumdiv(n,d,A048673(n/d)*A349358(d));

Formula

a(n) = Sum_{d|n} A048673(n/d) * A349358(d).