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A370904
Partial alternating sums of the sum of the bi-unitary divisors function (A188999).
1
1, -2, 2, -3, 3, -9, -1, -16, -6, -24, -12, -32, -18, -42, -18, -45, -27, -57, -37, -67, -35, -71, -47, -107, -81, -123, -83, -123, -93, -165, -133, -196, -148, -202, -154, -204, -166, -226, -170, -260, -218, -314, -270, -330, -270, -342, -294, -402, -352, -430
OFFSET
1,2
LINKS
László Tóth, Alternating Sums Concerning Multiplicative Arithmetic Functions, Journal of Integer Sequences, Vol. 20 (2017), Article 17.2.1.
FORMULA
a(n) = Sum_{k=1..n} (-1)^(k+1) * A188999(k).
a(n) = -(11/53) * c * n^2 + O(n * log(n)^3), where c = A307160 (Tóth, 2017).
MATHEMATICA
f[p_, e_] := If[OddQ[e], (p^(e+1)-1)/(p-1), (p^(e+1)-1)/(p-1)-p^(e/2)]; bsigma[1] = 1; bsigma[n_] := Times @@ f @@@ FactorInteger[n]; Accumulate[Array[(-1)^(# + 1) * bsigma[#] &, 100]]
PROG
(PARI) bsigma(n) = {my(f = factor(n)); prod(i=1, #f~, p = f[i, 1]; e = f[i, 2]; if(e%2, (p^(e+1)-1)/(p-1), (p^(e+1)-1)/(p-1)-p^(e/2))); }
lista(kmax) = {my(s = 0); for(k = 1, kmax, s += (-1)^(k+1) * bsigma(k); print1(s, ", "))};
CROSSREFS
Similar sequences: A068762, A068773, A307704, A357817, A362028.
Sequence in context: A236545 A321327 A370898 * A353563 A360250 A095060
KEYWORD
sign,easy
AUTHOR
Amiram Eldar, Mar 05 2024
STATUS
approved