A384057 The number of integers k from 1 to n such that the greatest divisor of k that is a unitary divisor of n is a 3-smooth number.

1, 2, 3, 4, 4, 6, 6, 8, 9, 8, 10, 12, 12, 12, 12, 16, 16, 18, 18, 16, 18, 20, 22, 24, 24, 24, 27, 24, 28, 24, 30, 32, 30, 32, 24, 36, 36, 36, 36, 32, 40, 36, 42, 40, 36, 44, 46, 48, 48, 48, 48, 48, 52, 54, 40, 48, 54, 56, 58, 48, 60, 60, 54, 64, 48, 60, 66, 64

Offset: 1

Author

Amiram Eldar, May 18 2025

Keywords

• nonn
• easy
• mult

Comments

First differs A372671 from at n = 25.

Links

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

Crossrefs

Unitary analog of A372671.

Cf. A003586, A065330, A065331, A065463, A372671, A384046, A384047.

The number of integers k from 1 to n such that the greatest divisor of k that is a unitary divisor of n is: A047994 (1), A384048 (squarefree), A384049 (cubefree), A384050 (powerful), A384051 (cubefull), A384052 (square), A384053 (cube), A384054 (exponentially odd), A384055 (odd), A384056 (power of 2), this sequence (3-smooth), A384058 (5-rough).

Programs

Mathematica

f[p_, e_] := p^e - If[p < 5, 0, 1]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a,100]

PARI

a(n) = {my(f = factor(n)); prod(i = 1, #f~, f[i,1]^f[i,2] - if(f[i,1] < 5, 0, 1));}

Formula

Multiplicative with a(p^e) = p^e if p <= 3, and p^e-1 if p >= 5.

a(n) = n * A047994(n) / A384058(n).

a(n) = A047994(A065330(n)) * A065331(n).

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

Sum_{k=1..n} a(k) ~ (36/55) * c * n^2, where c = Product_{p prime} (1 - 1/(p*(p+1))) = A065463.

In general, the average order of the number of integers k from 1 to n such that the greatest divisor of k that is a unitary divisor of n is a p-smooth number (i.e., not divisible by any prime larger than the prime p) is (1/2) * Product_{q prime <= p} (1 + 1/(q^2+q-1)) * A065463 * n^2.