A384058 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 5-rough number (A007310).

1, 1, 2, 3, 5, 2, 7, 7, 8, 5, 11, 6, 13, 7, 10, 15, 17, 8, 19, 15, 14, 11, 23, 14, 25, 13, 26, 21, 29, 10, 31, 31, 22, 17, 35, 24, 37, 19, 26, 35, 41, 14, 43, 33, 40, 23, 47, 30, 49, 25, 34, 39, 53, 26, 55, 49, 38, 29, 59, 30, 61, 31, 56, 63, 65, 22, 67, 51, 46

Offset

1,3

Links

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

Formula

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

a(n) = n * A047994(n) / A384057(n).

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

Dirichlet g.f.: zeta(s-1) * ((1 - 1/2^(s-1) + 1/2^(2*s-1))/(1 - 1/2^s)) * ((1 - 2/3^s + 1/3^(2*s-1))/(1 - 1/3^s)).

Sum_{k=1..n} a(k) ~ (55/144) * n^2.

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-rough number (i.e., not divisible by any prime smaller than the prime p) is (1/2) * Product_{q prime <= p} (1 - 1/q + 1/(q+1)) * n^2.

Mathematica

f[p_, e_] := p^e - If[p < 5, 1, 0]; 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, 1] < 5, 1, 0)); }

Crossrefs

Unitary analog of A384042.

Cf. A007310, A065330, A065331, 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), A384057 (3-smooth), this sequence (5-rough).

Sequence in context: A176195 A231233 A174562 * A224382 A069227 A117368

Adjacent sequences: A384055 A384056 A384057 * A384059 A384060 A384061

Keyword

nonn,easy,mult

Author

Amiram Eldar, May 18 2025

Status

approved