A357684 The squarefree part (A007913) of numbers whose squarefree part is a unitary divisor (A335275).

1, 2, 3, 1, 5, 6, 7, 1, 10, 11, 3, 13, 14, 15, 1, 17, 2, 19, 5, 21, 22, 23, 1, 26, 7, 29, 30, 31, 33, 34, 35, 1, 37, 38, 39, 41, 42, 43, 11, 5, 46, 47, 3, 1, 2, 51, 13, 53, 55, 57, 58, 59, 15, 61, 62, 7, 1, 65, 66, 67, 17, 69, 70, 71, 73, 74, 3, 19, 77, 78, 79

Offset

1,2

Links

Amiram Eldar, Table of n, a(n) for n = 1..10000

Rafael Jakimczuk, Two Topics in Number Theory: Sum of Divisors of the Primorial and Sum of Squarefree Parts, International Mathematical Forum, Vol. 12, No. 7 (2017), pp. 331-338.

Formula

a(n) = A007913(A335275(n)).

a(n) = 1 iff A335275(n) is a square (A000290).

a(n) = A335275(n) iff A335275(n) is squarefree (A005117).

Sum_{k, a(k) <= x} ~ c*x^2 + o(x^2), where c = (3/Pi^2) * Sum_{k>=1} f(k)/k^4 = 0.32103327852028541131..., and f(k) = Product_{p prime | k} (p/(p+1)) (Jakimczuk, 2017).

Sum_{k=1..n} a(k) ~ c'*x^2 + o(x^2), where c' = c / (A065465)^2 = 0.41313480468422995583... .

Mathematica

s[n_] := If[AllTrue[(f = FactorInteger[n])[[;; , 2]], # == 1 || EvenQ[#] &], i = Position[f[[;; , 2]], 1] // Flatten; Times @@ f[[i, 1]], Nothing]; Array[s, 100]

Prog

(PARI) s(n) = {my(f = factor(n), ans = 1); for(k = 1, #f~, if(f[k, 2] > 1 && f[k, 2]%2, ans = 0)); if(ans, ans = prod(k = 1, #f~, if(f[k, 2] == 1, f[k, 1], 1))) };
for(n = 1, 100, if(s(n) > 0, print1(s(n), ", ")))

Crossrefs

Cf. A000290, A005117, A007913, A008833, A065465, A069891, A335275.

Sequence in context: A160400 A325978 A326049 * A072400 A007913 A366244

Adjacent sequences: A357681 A357682 A357683 * A357685 A357686 A357687

Keyword

nonn

Author

Amiram Eldar, Oct 09 2022

Status

approved