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

1,1

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

Formula

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

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

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

Prog

(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(d<n, v[n/d]*u[d], 0)))); (u) }; \\ Compute the Dirichlet inverse of the sequence given in input vector v.
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));
v349348 = DirInverseCorrect(vector(up_to, n, A252463(n)));
A349348(n) = v349348[n];
A349349(n) = (A252463(n)+A349348(n));

Crossrefs

Coincides with A349126 on odd numbers.

Cf. A064989, A252463, A349348.

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

Sequence in context: A209915 A349386 A355687 * A349443 A319340 A349389

Adjacent sequences: A349346 A349347 A349348 * A349350 A349351 A349352

Keyword

nonn

Author

Antti Karttunen, Nov 15 2021

Status

approved