A082476 -id:A082476 - cp's OEIS frontend

A083399 Number of divisors of n that are not divisors of other divisors of n.

1, 2, 2, 2, 2, 3, 2, 2, 2, 3, 2, 3, 2, 3, 3, 2, 2, 3, 2, 3, 3, 3, 2, 3, 2, 3, 2, 3, 2, 4, 2, 2, 3, 3, 3, 3, 2, 3, 3, 3, 2, 4, 2, 3, 3, 3, 2, 3, 2, 3, 3, 3, 2, 3, 3, 3, 3, 3, 2, 4, 2, 3, 3, 2, 3, 4, 2, 3, 3, 4, 2, 3, 2, 3, 3, 3, 3, 4, 2, 3, 2, 3, 2, 4, 3, 3, 3, 3, 2, 4, 3, 3, 3, 3, 3, 3, 2, 3, 3, 3, 2, 4

Offset: 1

Reinhard Zumkeller, Jun 12 2003

a(n) <= tau(n); a(n) = tau(n) iff n is prime or n=1 (A008578, A000040); a(n)=tau(n)-1 iff n is semiprime (A001358).
Number of noncomposite divisors of n, (cf. A008578). - Jaroslav Krizek, Nov 25 2009
From Wilf A. Wilson, Jul 21 2017: (Start)
a(n) is the number of maximal subsemigroups of the annular Jones monoid of degree n.
a(n) is the number of maximal subsemigroups of the monoid of orientation-preserving mappings on a set with n elements.
a(n) + 1 is the number of maximal subsemigroups of the monoid of orientation-preserving partial mappings on a set with n elements.
(End)
This is the restricted growth sequence transform of A001221 (and thus also of A007875, A034444, A082476, A292586 and many other sequences). This follows from the formula a(n) = 1+A001221(n), and from the fact that for any n, A001221(n) <= 1+A001221(k) for all k = 1..(n-1). A067003 gives the ordinal transform of A001221. See also A292582, A292583, A292585. - Antti Karttunen, Sep 25 2017

			{1,2,3,4,6,8,12,24} are the divisors of n=24: 1, 2, 3, 4 and 6 divide not only 24, but also 8 or 12, therefore a(24) = 3.
{1,2,3,4,6,8,12,24} are the divisors of n=24: 1, 2 and 3 are noncomposites, therefore a(24) = 3. - _Jaroslav Krizek_, Nov 25 2009

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
James East, Jitender Kumar, James D. Mitchell, and Wilf A. Wilson, Maximal subsemigroups of finite transformation and partition monoids, arXiv:1706.04967 [math.GR], 2017. [_Wilf A. Wilson_, Jul 21 2017]

Cf. A000005, A001221, A067003, A055212, A077761.

a083399 = (+ 1) . a001221  -- Reinhard Zumkeller, Sep 14 2015

[(#(PrimeDivisors(n)))+1: n in [1..100]]; // Vincenzo Librandi, Feb 15 2015

A083399 := proc(n)
    1+nops(numtheory[factorset](n)) ;
end proc:
seq(A083399(n),n=1..100) ; # R. J. Mathar, Sep 26 2017

A083399[n_Integer]:=1+PrimeNu[n]; (* Enrique Pérez Herrero, Jun 25 2010 *)
Rest@ CoefficientList[ Series[(1/(1 - x)) + Sum[1/(1 - x^Prime[j]), {j, 200}], {x, 0, 111}], x] (* Robert G. Wilson v, Aug 16 2011 *)

a(n)=#factor(n)~+1 \\ Charles R Greathouse IV, Sep 14 2015

a(n) = omega(n) + 1, where omega = A001221.
a(n) = tau(n) - A055212(n) = A000005(n)-A055212(n).
a(n) = A000005(n) - A033273(n) + 1. - Jaroslav Krizek, Nov 25 2009
a(n) = A010553(A007947(n)) = A000005(A000005(A007947(n))) = tau_2(tau_2(rad(n))). - Enrique Pérez Herrero, Jun 25 2010
G.f.: x/(1 - x) + Sum_{k>=1} x^prime(k)/(1 - x^prime(k)). - Ilya Gutkovskiy, Mar 21 2017
Sum_{k=1..n} a(k) = n * (log(log(n)) + B + 1) + O(n/log(n)), where B is Mertens's constant (A077761). - Amiram Eldar, Sep 29 2024

A360996 Multiplicative with a(p^e) = 5*e, p prime and e > 0.

Original entry on oeis.org

1, 5, 5, 10, 5, 25, 5, 15, 10, 25, 5, 50, 5, 25, 25, 20, 5, 50, 5, 50, 25, 25, 5, 75, 10, 25, 15, 50, 5, 125, 5, 25, 25, 25, 25, 100, 5, 25, 25, 75, 5, 125, 5, 50, 50, 25, 5, 100, 10, 50, 25, 50, 5, 75, 25, 75, 25, 25, 5, 250, 5, 25, 50, 30, 25, 125, 5, 50, 25, 125, 5, 150

Offset: 1

Vaclav Kotesovec, Feb 28 2023

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

Cf. A005361 (multiplicative with a(p^e) = e), A000005 (e+1), A343443 (e+2), A360997 (e+3), A322327 (2*e), A048691 (2*e+1), A360908 (2*e-1), A226602 (3*e), A048785 (3*e+1), A360910 (3*e-1), A360909 (3*e+2), A360911 (3*e-2), A322328 (4*e).

Cf. A082476.

g[p_, e_] := 5*e; a[1] = 1; a[n_] := Times @@ g @@@ FactorInteger[n]; Array[a, 100]

for(n=1, 100, print1(direuler(p=2, n, (1+3*X+X^2)/(1-X)^2)[n], ", "))

Dirichlet g.f.: Product_{primes p} (1 + 5*p^s/(p^s - 1)^2).

a(n) = A005361(n) * A082476(n).

A365492 The number of divisors of the smallest 4th power divisible by n.

Original entry on oeis.org

1, 5, 5, 5, 5, 25, 5, 5, 5, 25, 5, 25, 5, 25, 25, 5, 5, 25, 5, 25, 25, 25, 5, 25, 5, 25, 5, 25, 5, 125, 5, 9, 25, 25, 25, 25, 5, 25, 25, 25, 5, 125, 5, 25, 25, 25, 5, 25, 5, 25, 25, 25, 5, 25, 25, 25, 25, 25, 5, 125, 5, 25, 25, 9, 25, 125, 5, 25, 25, 125, 5, 25

Offset: 1

Amiram Eldar, Sep 05 2023

First differs from A082476 at n = 32.

The number of divisors of the 4th root of the smallest 4th power divisible by n, A053166(n), is A365491(n).

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

Cf. A000005, A053166, A053167, A082476, A365491.

f[p_, e_] := 4*Ceiling[e/4] + 1; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

a(n) = vecprod(apply(x -> 4*((x-1)\4) + 5, factor(n)[, 2]));

a(n) = A000005(A053167(n)).
Multiplicative with a(p^e) = 4*ceiling(e/4) + 1.
Dirichlet g.f.: zeta(s) * zeta(4*s) * Product_{p prime} (1 + 4/p^s - 1/p^(4*s)).

A344328 Number of divisors of n^5.

Original entry on oeis.org

1, 6, 6, 11, 6, 36, 6, 16, 11, 36, 6, 66, 6, 36, 36, 21, 6, 66, 6, 66, 36, 36, 6, 96, 11, 36, 16, 66, 6, 216, 6, 26, 36, 36, 36, 121, 6, 36, 36, 96, 6, 216, 6, 66, 66, 36, 6, 126, 11, 66, 36, 66, 6, 96, 36, 96, 36, 36, 6, 396, 6, 36, 66, 31, 36, 216, 6, 66, 36, 216, 6, 176, 6, 36, 66, 66, 36

Offset: 1

Seiichi Manyama, May 15 2021

Seiichi Manyama, Table of n, a(n) for n = 1..10000

Column k=5 of A343656.

Cf. A000005, A000584, A082476 (5^omega(n)), A203556.

Table[DivisorSigma[0, n^5], {n, 1, 100}] (* Amiram Eldar, May 15 2021 *)

PARI
```
a(n) = numdiv(n^5);
```

a(n) = prod(k=1, #f=factor(n)[, 2], 5*f[k]+1);

PARI
```
a(n) = sumdiv(n, d, 5^omega(d));
```

my(N=99, x='x+O('x^N)); Vec(sum(k=1, N, 5^omega(k)*x^k/(1-x^k)))

for(n=1, 100, print1(direuler(p=2, n, (1 + 4*X)/(1 - X)^2)[n], ", ")) \\ Vaclav Kotesovec, Aug 19 2021

a(n) = A000005(A000584(n)).
Multiplicative with a(p^e) = 5*e+1.
a(n) = Sum_{d|n} 5^omega(d).
G.f.: Sum_{k>=1} 5^omega(k) * x^k/(1 - x^k).
Dirichlet g.f.: zeta(s)^2 * Product_{primes p} (1 + 4/p^s). - Vaclav Kotesovec, Aug 19 2021

A344337 a(n) = 9^omega(n), where omega(n) is the number of distinct primes dividing n.

Original entry on oeis.org

1, 9, 9, 9, 9, 81, 9, 9, 9, 81, 9, 81, 9, 81, 81, 9, 9, 81, 9, 81, 81, 81, 9, 81, 9, 81, 9, 81, 9, 729, 9, 9, 81, 81, 81, 81, 9, 81, 81, 81, 9, 729, 9, 81, 81, 81, 9, 81, 9, 81, 81, 81, 9, 81, 81, 81, 81, 81, 9, 729, 9, 81, 81, 9, 81, 729, 9, 81, 81, 729, 9, 81, 9, 81, 81, 81

Offset: 1

Seiichi Manyama, May 15 2021

k^omega(n): A034444 (k=2), A074816 (k=3), A082476 (k=5), this sequence (k=9).

Cf. A000005, A001019, A001221.

Table[9^PrimeNu[n], {n, 1, 100}] (* Amiram Eldar, May 15 2021 *)

PARI
```
a(n) = 9^omega(n);
```

a(n) = prod(k=1, #f=factor(n)[, 2], 9);

a(n) = sumdiv(n, d, moebius(d)^2*numdiv(d)^3);

a(n) = A001019(A001221(n)).
Multiplicative with a(p^e) = 9.
a(n) = Sum_{d|n} mu(d)^2 * tau(d)^3.
Dirichlet g.f.: Product_{p prime} (1 + 9/(p^s-1)). - Amiram Eldar, Sep 19 2023

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A083399 Number of divisors of n that are not divisors of other divisors of n.

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A360996 Multiplicative with a(p^e) = 5*e, p prime and e > 0.

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A365492 The number of divisors of the smallest 4th power divisible by n.

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A344328 Number of divisors of n^5.

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A344337 a(n) = 9^omega(n), where omega(n) is the number of distinct primes dividing n.

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