A349438 Dirichlet convolution of A000027 with A349348 (Dirichlet inverse of A252463), where A252463 shifts the prime factorization of odd numbers one step towards smaller primes and divides even numbers by two.

1, 1, 1, 1, 2, 0, 2, 1, 3, 0, 4, -1, 2, 0, 2, 1, 4, -1, 2, -2, 2, 0, 4, -2, 10, 0, 9, -2, 6, -4, 2, 1, 4, 0, 4, -4, 6, 0, 2, -4, 4, -4, 2, -4, 6, 0, 4, -3, 14, -4, 4, -2, 6, -6, 8, -4, 2, 0, 6, -6, 2, 0, 6, 1, 4, -8, 6, -4, 4, -8, 4, -6, 2, 0, 10, -2, 8, -4, 6, -6, 27, 0, 4, -6, 8, 0, 6, -8, 6, -16, 4, -4, 2, 0, 4

Offset

1,5

Comments

Convolving this sequence with A348045 gives Euler phi, A000010.

It might first seem that A000265(a(p^k)) = p^(k-1) for all odd primes and all exponents k >= 1, but this does not hold for prime 37. However, with p=37, identity A065330(A349438(37^k)) = 37^(k-1) seems to hold for all exponents k >= 1. - Antti Karttunen, Nov 20 2021

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} d * A349348(n/d).

Mathematica

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

Prog

(PARI)
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));
memoA349348 = Map();
A349348(n) = if(1==n, 1, my(v); if(mapisdefined(memoA349348, n, &v), v, v = -sumdiv(n, d, if(d<n, A252463(n/d)*A349348(d), 0)); mapput(memoA349348, n, v); (v)));
A349438(n) = sumdiv(n, d, d*A349348(n/d));

Crossrefs

Cf. A000027, A064989, A252463, A349348, A349437 (Dirichlet inverse), A349439 (sum with it).

Cf. also A000010, A000265, A065330, A348045.

Sequence in context: A074397 A345039 A082023 * A078152 A339242 A339222

Adjacent sequences: A349435 A349436 A349437 * A349439 A349440 A349441

Keyword

sign

Author

Antti Karttunen, Nov 18 2021

Status

approved