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

Author

Antti Karttunen, Nov 18 2021

Keywords

• sign

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

Crossrefs

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

Cf. also A000010, A000265, A065330, A348045.

Programs

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 *)

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(dA252463(n/d)*A349348(d),0)); mapput(memoA349348,n,v); (v)));
A349438(n) = sumdiv(n,d,d*A349348(n/d));

Formula

a(n) = Sum_{d|n} d * A349348(n/d).