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
Links
- Amiram Eldar, Table of n, a(n) for n = 1..10000
Crossrefs
Unitary analog of A384042.
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).
Programs
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Mathematica
f[p_, e_] := p^e - If[p < 5, 1, 0]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a,100]
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PARI
a(n) = {my(f = factor(n)); prod(i = 1, #f~, f[i,1]^f[i,2] - if(f[i,1] < 5, 1, 0));}
Formula
Multiplicative with a(p^e) = p^e-1 if p <= 3, and p^e if p >= 5.
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.
Comments