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A349915
Dirichlet inverse of A113415, where A113415 is the arithmetic mean between the number and sum of the odd divisors of n.
6
1, -1, -3, 0, -4, 3, -5, 0, 1, 4, -7, 0, -8, 5, 10, 0, -10, -1, -11, 0, 12, 7, -13, 0, -1, 8, -1, 0, -16, -10, -17, 0, 16, 10, 14, 0, -20, 11, 18, 0, -22, -12, -23, 0, -2, 13, -25, 0, -5, 1, 22, 0, -28, 1, 18, 0, 24, 16, -31, 0, -32, 17, -2, 0, 20, -16, -35, 0, 28, -14, -37, 0, -38, 20, 5, 0, 20, -18, -41, 0, -2, 22
OFFSET
1,3
LINKS
FORMULA
a(1) = 1; a(n) = -Sum_{d|n, d < n} A113415(n/d) * a(d).
a(n) = A349916(n) - A113415(n).
MATHEMATICA
s[n_] := DivisorSum[n, (# + 1) * Mod[#, 2] &] / 2; a[1] = 1; a[n_] := a[n] = -DivisorSum[n, a[#] * s[n/#] &, # < n &]; Array[a, 100] (* Amiram Eldar, Dec 08 2021 *)
PROG
(PARI)
A113415(n) = if(n<1, 0, sumdiv(n, d, if(d%2, (d+1)/2)));
memoA349915 = Map();
A349915(n) = if(1==n, 1, my(v); if(mapisdefined(memoA349915, n, &v), v, v = -sumdiv(n, d, if(d<n, A113415(n/d)*A349915(d), 0)); mapput(memoA349915, n, v); (v)));
CROSSREFS
KEYWORD
sign
AUTHOR
Antti Karttunen, Dec 07 2021
STATUS
approved