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A323894
Sum of A048673 and its Dirichlet inverse, A323893.
8
2, 0, 0, 4, 0, 12, 0, 12, 9, 16, 0, 26, 0, 24, 24, 37, 0, 46, 0, 36, 36, 28, 0, 76, 16, 36, 51, 56, 0, 58, 0, 114, 42, 40, 48, 121, 0, 48, 54, 106, 0, 94, 0, 66, 104, 60, 0, 223, 36, 92, 60, 86, 0, 220, 56, 166, 72, 64, 0, 164, 0, 76, 162, 349, 72, 112, 0, 96, 90, 136, 0, 354, 0, 84, 150, 116, 84, 148, 0, 312, 277, 88, 0, 260, 80, 96, 96
OFFSET
1,1
COMMENTS
The first four negative terms are a(3063060) = -14126242, a(3423420) = -17546656, a(4084080) = -14460312, a(4144140) = -22677277. - Antti Karttunen, Apr 20 2022
FORMULA
a(n) = A048673(n) + A323893(n).
For n > 1, a(n) = -Sum_{d|n, 1<d<n} A048673(n/d) * A323893(d). - Antti Karttunen, Apr 20 2022
PROG
(PARI)
up_to = 65537;
DirInverse(v) = { my(u=vector(#v)); u[1] = (1/v[1]); for(n=2, #v, u[n] = -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.
A003961(n) = my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); \\ From A003961
A048673(n) = (A003961(n)+1)/2;
v323893 = DirInverse(vector(up_to, n, A048673(n)));
A323893(n) = v323893[n];
A323894(n) = (A048673(n)+A323893(n));
CROSSREFS
KEYWORD
sign
AUTHOR
Antti Karttunen, Feb 08 2019
STATUS
approved