cp's OEIS Frontend

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

%I A370903 #7 Mar 05 2024 11:50:50
%S A370903 1,0,1,-3,-2,-3,-2,-10,-1,-2,-1,-5,-4,-5,-4,-20,-19,-28,-27,-31,-30,
%T A370903 -31,-30,-38,-13,-14,13,9,10,9,10,-22,-21,-22,-21,-57,-56,-57,-56,-64,
%U A370903 -63,-64,-63,-67,-58,-59,-58,-74,-25,-50,-49,-53,-52,-79,-78,-86,-85
%N A370903 Partial alternating sums of the powerful part function (A057521).
%H A370903 Amiram Eldar, <a href="/A370903/b370903.txt">Table of n, a(n) for n = 1..10000</a>
%H A370903 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 A370903 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 A370903 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 A370903 (PARI) pfp(n) = {my(f = factor(n)); prod(i = 1, #f~, if(f[i, 2] == 1, 1, f[i, 1]^f[i, 2]));}
%o A370903 lista(kmax) = {my(s = 0); for(k = 1, kmax, s += (-1)^(k+1) * pfp(k); print1(s, ", "))};
%Y A370903 Cf. A057521, A370902.
%Y A370903 Cf. A002117, A013661, A078434.
%Y A370903 Similar sequences: A068762, A068773, A307704, A357817, A362028.
%K A370903 sign,easy
%O A370903 1,4
%A A370903 _Amiram Eldar_, Mar 05 2024