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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.
4
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
OFFSET
1,5
COMMENTS
Dirichlet convolution of this sequence with Euler phi (A000010) is A348045.
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
KEYWORD
sign
AUTHOR
Antti Karttunen, Nov 18 2021
STATUS
approved