A346234 Dirichlet inverse of A003961, where A003961(n) is fully multiplicative with a(prime(k)) = prime(k+1).

1, -3, -5, 0, -7, 15, -11, 0, 0, 21, -13, 0, -17, 33, 35, 0, -19, 0, -23, 0, 55, 39, -29, 0, 0, 51, 0, 0, -31, -105, -37, 0, 65, 57, 77, 0, -41, 69, 85, 0, -43, -165, -47, 0, 0, 87, -53, 0, 0, 0, 95, 0, -59, 0, 91, 0, 115, 93, -61, 0, -67, 111, 0, 0, 119, -195, -71, 0, 145, -231, -73, 0, -79, 123, 0, 0, 143, -255, -83, 0, 0, 129

Offset

1,2

Links

Antti Karttunen, Table of n, a(n) for n = 1..20000

Index entries for sequences computed from indices in prime factorization

Formula

a(n) = A055615(A003961(n)).

a(n) = A008683(n) * A003961(n).

Multiplicative with a(p^e) = 0 if e > 1, and -nextprime(p) otherwise, where nextprime function is A151800. - Antti Karttunen, Nov 14 2021

Prog

(PARI)
up_to = 16384;
DirInverseCorrect(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); };
v346234 = DirInverseCorrect(vector(up_to, n, A003961(n)));
A346234(n) = v346234[n];
(PARI)
A003961(n) = { my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
A346234(n) = (moebius(n)*A003961(n));
(PARI) A346234(n) = { my(f = factor(n)); prod(i=1, #f~, if(1<f[i, 2], 0, -nextprime(1+f[i, 1]))); }; \\ Antti Karttunen, Nov 14 2021

Crossrefs

Cf. A003961, A008683, A055615, A151800.

Cf. also A046692, A323893, A349125.

Sequence in context: A021885 A222480 A229984 * A346479 A346254 A021289

Adjacent sequences: A346231 A346232 A346233 * A346235 A346236 A346237

Keyword

sign,mult

Author

Antti Karttunen, Jul 11 2021

Extensions

Keyword:mult added by Antti Karttunen, Nov 14 2021

Status

approved