A384049 The number of integers k from 1 to n such that the greatest divisor of k that is a unitary divisor of n is cubefree.

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

Offset

1,2

Links

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

Formula

Multiplicative with a(p^e) = p^e if e <= 2, and p^e - 1 if e >= 3.

a(n) = n * A047994(n) / A384051(n).

a(n) = A047994(A360540(n)) * A360539(n).

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

Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = Product_{p prime} (1 - 1/(p^5*(p+1))) = 0.988504... (A065468).

Mathematica

f[p_, e_] := p^e - If[e < 3, 0, 1]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

Prog

(PARI) a(n) = {my(f = factor(n)); prod(i = 1, #f~, f[i, 1]^f[i, 2] - if(f[i, 2] < 3, 0, 1)); }

Crossrefs

Unitary analog of A254926.

Cf. A004709, A065468, A360539, A360540, 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), this sequence (cubefree), A384050 (powerful), A384051 (cubefull), A384052 (square), A384053 (cube), A384054 (exponentially odd), A384055 (odd), A384056 (power of 2), A384057 (3-smooth), A384058 (5-rough).

Sequence in context: A265539 A254926 A155919 * A265540 A075011 A283001

Adjacent sequences: A384046 A384047 A384048 * A384050 A384051 A384052

Keyword

nonn,easy,mult

Author

Amiram Eldar, May 18 2025

Status

approved