A349437 - OEIS

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A349437

Dirichlet convolution of A252463 with A055615 (Dirichlet inverse of n), where A252463 shifts the prime factorization of odd numbers one step towards smaller primes and divides even numbers by two.

1, -1, -1, 0, -2, 2, -2, 0, -2, 4, -4, 0, -2, 4, 2, 0, -4, 4, -2, 0, 2, 8, -4, 0, -6, 4, -4, 0, -6, -4, -2, 0, 4, 8, 4, 0, -6, 4, 2, 0, -4, -4, -2, 0, 4, 8, -4, 0, -10, 12, 4, 0, -6, 8, 8, 0, 2, 12, -6, 0, -2, 4, 4, 0, 4, -8, -6, 0, 4, -8, -4, 0, -2, 12, 6, 0, 8, -4, -6, 0, -8, 8, -4, 0, 8, 4, 6, 0, -6, -8, 4, 0, 2

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OFFSET

1,5

COMMENTS

Dirichlet convolution of this sequence with Euler phi (A000010) is A348045.

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) = Sum_{d|n} A055615(n/d) * A252463(d).

MATHEMATICA

f[p_, e_] := NextPrime[p, -1]^e; s[1] = 1; s[n_] := If[EvenQ[n], n/2, Times @@ f @@@ FactorInteger[n]]; a[n_] := DivisorSum[n, # * MoebiusMu[#] * s[n/#] &]; Array[a, 100] (* Amiram Eldar, Nov 18 2021 *)

PROG

(PARI)

A055615(n) = (n*moebius(n));

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)};

A252463(n) = if(!(n%2), n/2, A064989(n));

A349437(n) = sumdiv(n, d, A055615(n/d)*A252463(d));

CROSSREFS

Cf. A055615, A064989, A252463, A349438 (Dirichlet inverse), A349439 (sum with it).

Cf. also A000010, A348045.

Sequence in context: A109135 A264136 A274850 * A215594 A230291 A338434

Adjacent sequences: A349434 A349435 A349436 * A349438 A349439 A349440

KEYWORD

sign

AUTHOR

Antti Karttunen, Nov 18 2021

STATUS

approved