%I #11 Mar 07 2024 01:26:14

%S 1,2,3,7,8,9,10,18,27,28,29,33,34,35,36,52,53,62,63,67,68,69,70,78,

%T 103,104,131,135,136,137,138,170,171,172,173,209,210,211,212,220,221,

%U 222,223,227,236,237,238,254,303,328,329,333,334,361,362,370,371,372,373

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

%H Amiram Eldar, <a href="/A370902/b370902.txt">Table of n, a(n) for n = 1..10000</a>

%H Maurice-Étienne Cloutier, <a href="http://hdl.handle.net/20.500.11794/28374">Les parties k-puissante et k-libre d’un nombre</a>, Thèse de doctorat, Université Laval (2018); <a href="https://dam-oclc.bac-lac.gc.ca/download?is_thesis=1&amp;oclc_number=1132187972&amp;wbdisable=true&amp;id=74dc2ac7-06b9-464c-9d86-0dc392701385&amp;fileName=33991.pdf">alternative link</a>.

%H Maurice-Étienne Cloutier, Jean-Marie De Koninck, and Nicolas Doyon, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL17/Cloutier/cloutier2.html">On the powerful and squarefree parts of an integer</a>, Journal of Integer Sequences, Vol. 17 (2014), Article 14.6.6.

%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} A057521(k).

%F a(n) = c_1 * n^(3/2) / 3 + c_2 * n^(4/3) / 4 + O(n^(6/5)), where c_1 = A328013 and c_2 are positive constants (Tóth, 2017).

%F c_2 = zeta(2/3) * Product_{p prime} (1 + 1/p^(4/3) - 2/p^2 - 1/p^(7/3) + 1/p^3) = -2.59305556147555965163... (László Tóth, personal communication). - _Amiram Eldar_, Mar 07 2024

%t f[p_, e_] := If[e == 1, 1, p^e]; pfp[n_] := Times @@ f @@@ FactorInteger[n]; pfp[1] = 1; Accumulate[Array[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 += pfp(k); print1(s, ", "))};

%Y Cf. A057521, A328013, A370903.

%K nonn,easy

%O 1,2

%A _Amiram Eldar_, Mar 05 2024