A347098 a(1) = 1; a(n) = -Sum_{d|n, d < n} A336853(n/d) * a(d), where A336853(n) = A003961(n) - n.

1, -1, -2, -4, -2, -5, -4, -10, -12, -7, -2, -1, -4, -11, -12, -16, -2, -1, -4, -7, -18, -13, -6, 42, -20, -17, -42, -5, -2, 21, -6, -4, -24, -19, -26, 106, -4, -23, -30, 38, -2, 45, -4, -25, -10, -29, -6, 196, -56, -17, -36, -23, -6, 123, -28, 82, -42, -31, -2, 225, -6, -37, 4, 80, -38, 15, -4, -43, -52, 39, -2, 413

Offset

1,3

Comments

Dirichlet inverse of the pointwise sum of A336853 and A063524 (1, 0, 0, 0, ...).

Links

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

Index entries for primes, gaps between

Index entries for sequences computed from indices in prime factorization

Formula

a(1) = 1; and for n > 1, a(n) = -Sum_{d|n, d < n} A336853(n/d) * a(d).

For all n >= 1, a(A000040(n)) = -A001223(n).

Mathematica

f[p_, e_] := NextPrime[p]^e; s[1] = 0; s[n_] := Times @@ f @@@ FactorInteger[n] - n; a[1] = 1; a[n_] := a[n] = -DivisorSum[n, a[#] * s[n/#] &, # < n &]; Array[a, 100] (* Amiram Eldar, Nov 27 2021 *)

Prog

(PARI)
up_to = 16384;
A336853(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); (factorback(f)-n); };
Aux347098(n) = if(1==n, n, A336853(n));
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.
v347098 = DirInverseCorrect(vector(up_to, n, Aux347098(n)));
A347098(n) = v347098[n];

Crossrefs

Cf. A000040, A001223, A003961, A336853, A347099, A347100.

Cf. also A346248, A347096.

Sequence in context: A366586 A020774 A291303 * A229920 A201562 A190041

Adjacent sequences: A347095 A347096 A347097 * A347099 A347100 A347101

Keyword

sign

Author

Antti Karttunen, Aug 19 2021

Status

approved