OFFSET
1,2
LINKS
Amiram Eldar, Table of n, a(n) for n = 1..10000
Maurice-Étienne Cloutier, Les parties k-puissante et k-libre d’un nombre, Thèse de doctorat, Université Laval (2018); alternative link.
Maurice-Étienne Cloutier, Jean-Marie De Koninck, and Nicolas Doyon, On the powerful and squarefree parts of an integer, Journal of Integer Sequences, Vol. 17 (2014), Article 14.6.6.
László Tóth, Alternating Sums Concerning Multiplicative Arithmetic Functions, Journal of Integer Sequences, Vol. 20 (2017), Article 17.2.1.
FORMULA
a(n) = Sum_{k=1..n} A057521(k).
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).
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
MATHEMATICA
f[p_, e_] := If[e == 1, 1, p^e]; pfp[n_] := Times @@ f @@@ FactorInteger[n]; pfp[1] = 1; Accumulate[Array[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 += pfp(k); print1(s, ", "))};
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Amiram Eldar, Mar 05 2024
STATUS
approved