A349349 Sum of A252463 and its Dirichlet inverse, where A252463 shifts the prime factorization of odd numbers one step towards smaller primes and divides even numbers by two.

2, 0, 0, 1, 0, 4, 0, 3, 4, 6, 0, 8, 0, 10, 12, 7, 0, 8, 0, 13, 20, 14, 0, 15, 9, 22, 8, 19, 0, 14, 0, 15, 28, 26, 30, 19, 0, 34, 44, 25, 0, 18, 0, 29, 12, 38, 0, 28, 25, 21, 52, 37, 0, 24, 42, 35, 68, 46, 0, 28, 0, 58, 20, 31, 66, 30, 0, 47, 76, 32, 0, 38, 0, 62, 18, 55, 70, 30, 0, 47, 16, 74, 0, 36, 78, 82, 92, 55

Offset: 1

Author

Antti Karttunen, Nov 15 2021

Keywords

• nonn

Comments

Question: Are there any negative terms? All terms in range 1 .. 2^23 are nonnegative. (See also A349126). - Antti Karttunen, Apr 20 2022

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 A349126 on odd numbers.

Cf. A064989, A252463, A349348.

Cf. also A323365, A323412, A323894, A349135, A353336.

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];
A349349(n) = (A252463(n)+A349348(n));

Formula

a(n) = A252463(n) + A349348(n).

a(1) = 2, and for n > 1, a(n) = -Sum_{d|n, 1A252463(d) * A349348(n/d).

For all n >= 1, a(2n-1) = A349126(2n-1).