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

1,2

Links

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

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... .

Mathematica

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

Prog

(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

Crossrefs

Cf. A062503, A337050, A368884.

Sequence in context: A214392 A071975 A350389 * A182659 A359593 A358881

Adjacent sequences: A368883 A368884 A368885 * A368887 A368888 A368889

Keyword

nonn,easy,mult

Author

Amiram Eldar, Jan 09 2024

Status

approved