A349626 Möbius transform of A326042, where A326042(n) = A064989(sigma(A003961(n))).

1, 0, 1, 10, 0, 0, 1, -8, 27, 0, 4, 10, 3, 0, 0, 46, 2, 0, 1, 0, 1, 0, 5, -8, 33, 0, -7, 10, 0, 0, 16, 6, 4, 0, 0, 270, 9, 0, 3, 0, 6, 0, 1, 40, 0, 0, 7, 46, 83, 0, 2, 30, 5, 0, 0, -8, 1, 0, 28, 0, 12, 0, 27, 1036, 0, 0, 3, 20, 5, 0, 30, -216, 2, 0, 33, 10, 4, 0, 9, 0, 447, 0, 11, 10, 0, 0, 0, -32, 24, 0, 3, 50, 16

Offset

1,4

Comments

Dirichlet convolution of Euler phi (A000010) with A349624.

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(n) = Sum_{d|n} A008683(n/d) * A326042(d).

a(n) = Sum_{d|n} A000010(n/d) * A349624(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[n_] := DivisorSum[n, s[#] * MoebiusMu[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)));
A349626(n) = sumdiv(n, d, moebius(n/d)*A326042(d));

Crossrefs

Cf. A000010, A003961, A008683, A064989, A326042, A349623, A349624.

Sequence in context: A063699 A287464 A288012 * A167165 A288435 A287734

Adjacent sequences: A349623 A349624 A349625 * A349627 A349628 A349629

Keyword

sign,mult

Author

Antti Karttunen, Nov 26 2021

Status

approved