A368886 The largest unitary divisor of n without an exponent 2 in its prime factorization (A337050).

1, 2, 3, 1, 5, 6, 7, 8, 1, 10, 11, 3, 13, 14, 15, 16, 17, 2, 19, 5, 21, 22, 23, 24, 1, 26, 27, 7, 29, 30, 31, 32, 33, 34, 35, 1, 37, 38, 39, 40, 41, 42, 43, 11, 5, 46, 47, 48, 1, 2, 51, 13, 53, 54, 55, 56, 57, 58, 59, 15, 61, 62, 7, 64, 65, 66, 67, 17, 69, 70

Offset: 1

Author

Amiram Eldar, Jan 09 2024

Keywords

• nonn
• easy
• mult

Links

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

Crossrefs

Cf. A062503, A337050, A368884.

Programs

Mathematica

f[p_, e_] := If[e == 2, 1, p^e]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

PARI

a(n) = {my(f = factor(n)); prod(i = 1, #f~, if(f[i,2] == 2, 1, f[i,1]^f[i,2]));}

Python

from math import prod
from sympy import factorint
def A368886(n): return prod(p**e for p, e in factorint(n).items() if e!=2) # Chai Wah Wu, Jan 09 2024

Formula

Multiplicative with a(p^2) = 1, and a(p^e) = p^e if e != 2.

a(n) = n / A368884(n).

a(n) >= 1, with equality if and only if n is in A062503.

a(n) <= n, with equality if and only if n is in A337050.

Dirichlet g.f.: zeta(s-1) * Product_{p prime} (1 - 1/p^(2*s-2) + 1/p^(2*s) + 1/p^(3*s-3) - 1/p^(3*s-1)).

Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = Product_{p prime} (1 - 1/p^2 + 1/p^3 + 1/p^4 - 1/p^5) = 0.78357388280736936739... .