A379714 Partial alternating sums of the number of exponential divisors function (A049419).

1, 0, 1, -1, 0, -1, 0, -2, 0, -1, 0, -2, -1, -2, -1, -4, -3, -5, -4, -6, -5, -6, -5, -7, -5, -6, -4, -6, -5, -6, -5, -7, -6, -7, -6, -10, -9, -10, -9, -11, -10, -11, -10, -12, -10, -11, -10, -13, -11, -13, -12, -14, -13, -15, -14, -16, -15, -16, -15, -17, -16

Offset: 1

Author

Amiram Eldar, Dec 30 2024

Keywords

• sign
• easy

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000
• László Tóth, Alternating Sums Concerning Multiplicative Arithmetic Functions, Journal of Integer Sequences, Vol. 20 (2017), Article 17.2.1. See section 4.15, p. 36.

Crossrefs

Cf. A049419, A065442, A145353, A327837.

Similar sequences: A068762, A068773, A307704, A357817, A370895, A370896.

Programs

Mathematica

f[p_, e_]  := DivisorSigma[0, e]; ediv[n_] := Times @@ f @@@ FactorInteger[n]; Accumulate[Table[(-1)^(n+1)*ediv[n], {n, 1, 100}]]

PARI

ediv(n) = vecprod(apply(numdiv, factor(n)[, 2]));
list(nmax) = {my(s = 0); for(k = 1, nmax, s += (-1)^(k+1) * ediv(k); print1(s, ", "))};

Formula

a(n) = Sum_{k=1..n} (-1)^(k+1) * A049419(k).

Limit_{n->oo} a(n)/n = A327837 * (2/(A065442 + 1) - 1) = -0.37293122584744001729... .