VOOZH about

URL: https://oeis.org/A384057

⇱ A384057 - OEIS


login
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.
16
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
1,2
COMMENTS
First differs A372671 from at n = 25.
LINKS
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.
MATHEMATICA
f[p_, e_] := p^e - If[p < 5, 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, 1] < 5, 0, 1)); }
CROSSREFS
Unitary analog of A372671.
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).
Sequence in context: A102441 A102440 A372671 * A124815 A081328 A179276
KEYWORD
nonn,easy,mult
AUTHOR
Amiram Eldar, May 18 2025
STATUS
approved