A370895 - OEIS

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A370895

Partial alternating sums of Pillai's arithmetical function (A018804).

1, -2, 3, -5, 4, -11, 2, -18, 3, -24, -3, -43, -18, -57, -12, -60, -27, -90, -53, -125, -60, -123, -78, -178, -113, -188, -107, -211, -154, -289, -228, -340, -235, -334, -217, -385, -312, -423, -298, -478, -397, -592, -507, -675, -486, -621, -528, -768, -635, -830

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OFFSET

1,2

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.

FORMULA

a(n) = Sum_{k=1..n} (-1)^(k+1) * A018804(k).

a(n) = -(1/Pi^2) * n^2 * (log(n) + 2*gamma - 1/2 - zeta'(2)/zeta(2) - 10*log(2)/3) + O(n^(547/416 + eps)), where gamma is Euler's constant (A001620) (Tóth, 2017).

MATHEMATICA

f[p_, e_] := (e*(p - 1)/p + 1)*p^e; pil[n_] := Times @@ f @@@ FactorInteger[n]; Accumulate[Array[(-1)^(#+1) * pil[#] &, 100]]

PROG

(PARI) pil(n) = {my(f=factor(n)); prod(i=1, #f~, (f[i, 2]*(f[i, 1]-1)/f[i, 1] + 1)*f[i, 1]^f[i, 2]); }

lista(kmax) = {my(s = 0); for(k = 1, kmax, s += (-1)^(k+1) * pil(k); print1(s, ", "))};

CROSSREFS

Cf. A018804, A272718.

Cf. A001620, A306016.

Similar sequences: A068762, A068773, A307704, A357817, A362028.

Sequence in context: A227913 A079521 A325549 * A112060 A084933 A355065

Adjacent sequences: A370892 A370893 A370894 * A370896 A370897 A370898

KEYWORD

sign,easy

AUTHOR

Amiram Eldar, Mar 05 2024

STATUS

approved