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Partial alternating sums of the powerful part function (A057521).
2

%I #7 Mar 05 2024 11:50:50

%S 1,0,1,-3,-2,-3,-2,-10,-1,-2,-1,-5,-4,-5,-4,-20,-19,-28,-27,-31,-30,

%T -31,-30,-38,-13,-14,13,9,10,9,10,-22,-21,-22,-21,-57,-56,-57,-56,-64,

%U -63,-64,-63,-67,-58,-59,-58,-74,-25,-50,-49,-53,-52,-79,-78,-86,-85

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

%H Amiram Eldar, <a href="/A370903/b370903.txt">Table of n, a(n) for n = 1..10000</a>

%H László Tóth, <a href="https://www.emis.de/journals/JIS/VOL20/Toth/toth25.html">Alternating Sums Concerning Multiplicative Arithmetic Functions</a>, Journal of Integer Sequences, Vol. 20 (2017), Article 17.2.1.

%F 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).

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

%o (PARI) pfp(n) = {my(f = factor(n)); prod(i = 1, #f~, if(f[i, 2] == 1, 1, f[i, 1]^f[i, 2]));}

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

%Y Cf. A057521, A370902.

%Y Cf. A002117, A013661, A078434.

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

%K sign,easy

%O 1,4

%A _Amiram Eldar_, Mar 05 2024