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, -2, -1, -3, 1, -5, -1, 0, 1, -7, 2, -11, 3, 6, -1, -13, -1, -17, 3, 10, 3, -19, 3, 0, 9, 0, 5, -23, -1, -29, -1, 14, 9, 15, 1, -31, 15, 22, 5, -37, -3, -41, 7, 0, 15, -43, 4, 0, -4, 26, 11, -47, -3, 21, 7, 34, 17, -53, -2, -59, 27, 0, -1, 33, -3, -61, 13, 38, -3, -67, 2, -71, 25, 0, 17, 35, -9, -73, 7, 0, 33

Offset: 1

Author

Antti Karttunen, Nov 15 2021

Keywords

• sign

Links

• Antti Karttunen, Table of n, a(n) for n = 1..20000
• Index entries for sequences computed from indices in prime factorization

Crossrefs

Coincides with A349125 on odd numbers.

Cf. A064989, A252463, A349349.

Cf. also A348045, A349437, A349438.

Programs

PARI

up_to = 20000;
DirInverseCorrect(v) = { my(u=vector(#v)); u[1] = (1/v[1]); for(n=2, #v, u[n] = (-u[1]*sumdiv(n, d, if(dA064989(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));
v349348 = DirInverseCorrect(vector(up_to,n,A252463(n)));
A349348(n) = v349348[n];

Formula

a(1) = 1; a(n) = -Sum_{d|n, d < n} A252463(n/d) * a(d).

a(n) = A349349(n) - A252463(n).

For all n >= 1, a(2n-1) = A349125(2n-1).