A385007 The largest unitary divisor of n that is a biquadratefree number (A046100).

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 1, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 1, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 3, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 1, 65, 66, 67, 68, 69

Offset: 1

Author

Amiram Eldar, Jun 15 2025

Keywords

• nonn
• easy
• mult

Comments

First differs from A053165 at n = 32 = 2^5: a(32) = 1 while A053165(32) = 2.

First differs from A383764 at n = 32 = 2^5: a(32) = 1 while A383764(32) = 32.

Equivalently, a(n) is the least divisor d of n such that n/d is a 4-full number (A036967).

Links

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

Crossrefs

Cf. A013661, A036967, A046100, A053165, A058035, A077610, A383764.

The largest unitary divisor of n that is: A000265 (odd), A006519 (power of 2), A055231 (squarefree), A057521 (powerful), A065330 (5-rough), A065331 (3-smooth), A350388 (square), A350389 (exponentially odd), A360539 (cubefree), A360540 (cubefull), A366126 (cube), A367168 (exponentially 2^n), this sequence (biquadratefree).

Programs

Mathematica

f[p_, e_] := If[e < 4, p^e, 1]; 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] < 4, f[i, 1]^f[i, 2], 1)); }

Formula

a(n) = 1 if and only if n is a 4-full number (A036967).

a(n) = n if and only if n is a biquadratefree number (A046100).

Multiplicative with a(p^e) = p^e if e <= 3, and 1 otherwise.

Dirichlet g.f.: zeta(s) * Product_{p prime} (1 + p^(1-s) - p^(-s) + p^(2-2*s) - p^(1-2*s) - p^(2-3*s) + p^(3-3*s) - p^(3-4*s) + p^(-4*s)).

Sum_{k=1..n} a(k) ~ c * zeta(2) * n^2 / 2, where c = Product_{p prime} (1 - 1/p^2 - 1/p^4 + 1/p^6 + 1/p^8 - 1/p^9) = 0.56331392082909224894... .