A349623 Dirichlet inverse of A326042, where A326042(n) = A064989(sigma(A003961(n))).

1, -1, -2, -10, -1, 2, -2, 18, -25, 1, -5, 20, -4, 2, 2, 46, -3, 25, -2, 10, 4, 5, -6, -36, -33, 4, 86, 20, -1, -2, -17, -220, 10, 3, 2, 250, -10, 2, 8, -18, -7, -4, -2, 50, 25, 6, -8, -92, -81, 33, 6, 40, -6, -86, 5, -36, 4, 1, -29, -20, -13, 17, 50, -886, 4, -10, -4, 30, 12, -2, -31, -450, -3, 10, 66, 20, 10, -8, -10

Offset

1,3

Comments

Multiplicative because A326042 is.

Links

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

Index entries for sequences computed from indices in prime factorization

Index entries for sequences related to sigma(n)

Formula

a(1) = 1; a(n) = -Sum_{d|n, d < n} A326042(n/d) * a(d).

Mathematica

f1[p_, e_] := NextPrime[p]^e; s1[1] = 1; s1[n_] := Times @@ f1 @@@ FactorInteger[n]; f2[2, e_] := 1; f2[p_, e_] := NextPrime[p, -1]^e; s2[1] = 1; s2[n_] := Times @@ f2 @@@ FactorInteger[n]; s[n_] := s2[DivisorSigma[1, s1[n]]]; a[1] = 1; a[n_] := a[n] = -DivisorSum[n, a[#]*s[n/#] &, # < n &]; Array[a, 100] (* Amiram Eldar, Nov 27 2021 *)

Prog

(PARI)
A003961(n) = my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); \\ From A003961
A064989(n) = {my(f); f = factor(n); 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)};
A326042(n) = A064989(sigma(A003961(n)));
memoA349623 = Map();
A349623(n) = if(1==n, 1, my(v); if(mapisdefined(memoA349623, n, &v), v, v = -sumdiv(n, d, if(d<n, A326042(n/d)*A349623(d), 0)); mapput(memoA349623, n, v); (v)));

Crossrefs

Cf. A000203, A003961, A064989, A326042, A349624, A349625, A349626.

Sequence in context: A189871 A096877 A058297 * A113160 A100078 A051242

Adjacent sequences: A349620 A349621 A349622 * A349624 A349625 A349626

Keyword

sign,mult

Author

Antti Karttunen, Nov 26 2021

Status

approved