A323893 Dirichlet inverse of A048673, where A048673(n) = (A003961(n)+1) / 2, and A003961 is fully multiplicative with a(prime(i)) = prime(i+1).

1, -2, -3, -1, -4, 4, -6, -2, -4, 5, -7, 3, -9, 7, 6, -4, -10, 8, -12, 4, 8, 8, -15, 8, -9, 10, -12, 6, -16, 5, -19, -8, 9, 11, 9, 8, -21, 13, 11, 11, -22, 11, -24, 7, 16, 16, -27, 20, -25, 18, 12, 9, -30, 32, 10, 17, 14, 17, -31, 6, -34, 20, 24, -16, 12, 14, -36, 10, 17, 20, -37, 16, -40, 22, 27, 12, 12, 20, -42, 28, -36, 23, -45, 12, 13

Offset

1,2

Links

Antti Karttunen, Table of n, a(n) for n = 1..16383

Index entries for sequences related to prime indices in the factorization of n.

Formula

a(1) = 1, and for n > 1, a(n) = -Sum_{d|n, d<n} A048673(n/d) * a(d).

a(n) = A349134(A003961(n)). - Antti Karttunen, Nov 30 2024

Prog

(PARI)
up_to = 20000;
DirInverse(v) = { my(u=vector(#v)); u[1] = (1/v[1]); for(n=2, #v, u[n] = (-u[1]*sumdiv(n, d, if(d<n, v[n/d]*u[d], 0)))); (u) }; \\ Compute the Dirichlet inverse of the sequence given in input vector v.
A003961(n) = my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); \\ From A003961
A048673(n) = (A003961(n)+1)/2;
v323893 = DirInverse(vector(up_to, n, A048673(n)));
A323893(n) = v323893[n];
(PARI)
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))); \\ Antti Karttunen, Nov 30 2024

Crossrefs

Cf. A003961, A048673, A323894, A349134, A378520 (Möbius transform).

Sequence in context: A324752 A298637 A034867 * A329721 A193790 A055446

Adjacent sequences: A323890 A323891 A323892 * A323894 A323895 A323896

Keyword

sign

Author

Antti Karttunen, Feb 08 2019

Status

approved