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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.
4
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
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.
Sequence in context: A209915 A349386 A355687 * A349443 A319340 A349389
KEYWORD
nonn
AUTHOR
Antti Karttunen, Nov 15 2021
STATUS
approved