A370902 Partial sums of the powerful part function (A057521).
1, 2, 3, 7, 8, 9, 10, 18, 27, 28, 29, 33, 34, 35, 36, 52, 53, 62, 63, 67, 68, 69, 70, 78, 103, 104, 131, 135, 136, 137, 138, 170, 171, 172, 173, 209, 210, 211, 212, 220, 221, 222, 223, 227, 236, 237, 238, 254, 303, 328, 329, 333, 334, 361, 362, 370, 371, 372, 373
Offset: 1
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.
Programs
-
Mathematica
f[p_, e_] := If[e == 1, 1, p^e]; pfp[n_] := Times @@ f @@@ FactorInteger[n]; pfp[1] = 1; Accumulate[Array[pfp[#] &, 100]]
-
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, ", "))};
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