A370903 Partial alternating sums of the powerful part function (A057521).

1, 0, 1, -3, -2, -3, -2, -10, -1, -2, -1, -5, -4, -5, -4, -20, -19, -28, -27, -31, -30, -31, -30, -38, -13, -14, 13, 9, 10, 9, 10, -22, -21, -22, -21, -57, -56, -57, -56, -64, -63, -64, -63, -67, -58, -59, -58, -74, -25, -50, -49, -53, -52, -79, -78, -86, -85

Offset

1,4

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) = c_1 * n^(3/2) + c_2 * n^(4/3) + O(n^(6/5)), where c_1 = (zeta(3/2)/(3*zeta(3))) * ((9-12*sqrt(2))/23) * Product_{p prime} (1 + (sqrt(p)-1)/(p*(p-sqrt(p)+1))) = -0.40656281796860400941..., and c_2 = (zeta(4/3)/(4*zeta(2))) * ((2^(5/3)-3*2^(1/3)-1)/(2^(5/3)-2^(1/3)+1)) * Product_{p prime} (1 + (p^(1/3)-1)/(p*(p^(2/3)-p^(1/3)+1))) = -0.52513876339565998938... (Tóth, 2017).

Mathematica

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

Prog

(PARI) pfp(n) = {my(f = factor(n)); prod(i = 1, #f~, if(f[i, 2] == 1, 1, f[i, 1]^f[i, 2])); }
lista(kmax) = {my(s = 0); for(k = 1, kmax, s += (-1)^(k+1) * pfp(k); print1(s, ", "))};

Crossrefs

Cf. A057521, A370902.

Cf. A002117, A013661, A078434.

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

Sequence in context: A230406 A362688 A214254 * A153092 A165601 A324030

Adjacent sequences: A370900 A370901 A370902 * A370904 A370905 A370906

Keyword

sign,easy

Author

Amiram Eldar, Mar 05 2024

Status

approved