A349397 Dirichlet convolution of A064216 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, 8, 50, -55, -8, 24, -6, 31, 15, -58, -20, 17, 31, -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

Offset

1,7

Comments

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

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

Links

Antti Karttunen, Table of n, a(n) for n = 1..16384

Index entries for sequences computed from indices in prime factorization

Formula

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

Prog

(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); };
memoA323893 = Map();
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)));
A349397(n) = sumdiv(n, d, A064216(n/d)*A323893(d));

Crossrefs

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

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

Sequence in context: A193505 A141848 A349398 * A140249 A335928 A081564

Adjacent sequences: A349394 A349395 A349396 * A349398 A349399 A349400

Keyword

sign

Author

Antti Karttunen, Nov 19 2021

Status

approved