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

1,7

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

Formula

a(n) = Sum_{d|n} A048673(n/d) * A349358(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); };
memoA349358 = Map();
A349358(n) = if(1==n, 1, my(v); if(mapisdefined(memoA349358, n, &v), v, v = -sumdiv(n, d, if(d<n, A064216(n/d)*A349358(d), 0)); mapput(memoA349358, n, v); (v)));
A349398(n) = sumdiv(n, d, A048673(n/d)*A349358(d));

Crossrefs

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

Cf. also A349376, A349377, A349385.

Sequence in context: A153420 A193505 A141848 * A349397 A140249 A335928

Adjacent sequences: A349395 A349396 A349397 * A349399 A349400 A349401

Keyword

sign

Author

Antti Karttunen, Nov 19 2021

Status

approved