A354347 Dirichlet inverse of A345000, where A345000(n) = gcd(A003415(n), A003415(A276086(n))), with A003415 the arithmetic derivative, and A276086 the primorial base exp-function.

1, -1, -1, -1, -1, 1, -1, -1, 0, 1, -1, 1, -1, 1, 1, -9, -1, -2, -1, 1, -3, 1, -1, 1, -4, -3, 0, 1, -1, -1, -1, 21, 1, 1, -1, -6, -1, 1, 1, 3, -1, 7, -1, -1, 0, -3, -1, 23, 0, 4, -3, 7, -1, 2, 1, 3, 1, 1, -1, -1, -1, 1, 8, 15, -1, -1, -1, 1, 1, 3, -1, 14, -1, 1, -46, -7, -1, 7, -1, 5, 0, 1, -1, 3, 1, -3, 1, -131

Offset

1,16

Links

Antti Karttunen, Table of n, a(n) for n = 1..30030

Index entries for sequences related to primorial base

Formula

a(1) = 1, and for n > 1, a(n) = -Sum_{d|n, d<n} A345000(n/d) * a(d).

For all n >= 1, A000035(a(n)) = A353627(n).

Prog

(PARI)
A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); };
A345000(n) = gcd(A003415(n), A003415(A276086(n)));
memoA354347 = Map();
A354347(n) = if(1==n, 1, my(v); if(mapisdefined(memoA354347, n, &v), v, v = -sumdiv(n, d, if(d<n, A345000(n/d)*A354347(d), 0)); mapput(memoA354347, n, v); (v)));

Crossrefs

Cf. A003415, A276086, A345000.

Cf. A038838 (positions of even terms), A122132 (of odd terms), A353627 (parity of terms), A354815 (positions of 0's), A354816 (of -1's).

Cf. also A346242, A354348, A354823, A354824.

Sequence in context: A021527 A257437 A339757 * A010166 A186116 A242627

Adjacent sequences: A354344 A354345 A354346 * A354348 A354349 A354350

Keyword

sign

Author

Antti Karttunen, Jun 07 2022

Status

approved