A370901 - OEIS

%I #8 Mar 05 2024 11:51:18

%S 1,-1,2,1,6,0,7,6,7,-3,8,5,18,4,19,18,35,33,52,47,68,46,69,66,67,41,

%T 42,35,64,34,65,64,97,63,98,97,134,96,135,130,171,129,172,161,166,120,

%U 167,164,165,163,214,201,254,252,307,300,357,299,358,343,404,342

%N Partial alternating sums of the powerfree part function (A055231).

%H Amiram Eldar, <a href="/A370901/b370901.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) = Sum_{k=1..n} (-1)^(k+1) * A055231(k).

%F a(n) = (5/38) * c * n^2 + O(R(n)), where c = Product_{p prime} (1 - (p^2+p-1)/(p^3*(p+1))) = 0.649606... (A191622), R(n) = x^(3/2) * exp(-c_1 * log(n)^(3/5) / log(log(n))^(1/5)) unconditionally, or x^(7/5) * exp(c_2 * log(n) / log(log(n))) assuming the Riemann hypothesis, and c_1 and c_2 are positive constants (Tóth, 2017).

%t f[p_, e_] := If[e == 1, p, 1]; 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, f[i, 1], 1));}

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

%Y Cf. A055231, A191622, A370900.

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

%K sign,easy

%O 1,3

%A _Amiram Eldar_, Mar 05 2024