A074816 -id:A074816 - cp's OEIS frontend

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

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

A375230 The total number of infinitary divisors of the infinitary divisors of n.

Original entry on oeis.org

1, 3, 3, 3, 3, 9, 3, 9, 3, 9, 3, 9, 3, 9, 9, 3, 3, 9, 3, 9, 9, 9, 3, 27, 3, 9, 9, 9, 3, 27, 3, 9, 9, 9, 9, 9, 3, 9, 9, 27, 3, 27, 3, 9, 9, 9, 3, 9, 3, 9, 9, 9, 3, 27, 9, 27, 9, 9, 3, 27, 3, 9, 9, 9, 9, 27, 3, 9, 9, 27, 3, 27, 3, 9, 9, 9, 9, 27, 3, 9, 3, 9, 3, 27

Offset: 1

Amiram Eldar, Aug 06 2024

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

Cf. A000120, A005117, A037445, A064547, A077609, A138302.

Similar sequences: A007425 (analogous with all the divisors), A074816 (unitary analog).

f[p_, e_] := 3^DigitCount[e, 2, 1]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

a(n) = vecprod(apply(x -> 3^hammingweight(x), factor(n)[,2]));

a(n) = Sum_{d infinitary divisor of n} A037445(d).
Multiplicative with a(p^e) = 3^A000120(e).
a(n) = 3^A064547(n).
a(n) = A007425(n) if and only if n is squarefree (A005117).
a(n) = A074816(n) if and only if n is in A138302.

A340227 Number of pairs of divisors of n, (d1,d2), such that d1 < d2 and d1*d2 is squarefree.

Original entry on oeis.org

0, 1, 1, 1, 1, 4, 1, 1, 1, 4, 1, 4, 1, 4, 4, 1, 1, 4, 1, 4, 4, 4, 1, 4, 1, 4, 1, 4, 1, 13, 1, 1, 4, 4, 4, 4, 1, 4, 4, 4, 1, 13, 1, 4, 4, 4, 1, 4, 1, 4, 4, 4, 1, 4, 4, 4, 4, 4, 1, 13, 1, 4, 4, 1, 4, 13, 1, 4, 4, 13, 1, 4, 1, 4, 4, 4, 4, 13, 1, 4, 1, 4, 1, 13, 4, 4, 4, 4, 1, 13

Offset: 1

Wesley Ivan Hurt, Jan 01 2021

nonn

If n = p where p is prime, the only pair of divisors of n such that d1 < d2 is (1,p). Since the product 1*p = p is squarefree, this satisfies the constraints. Thus, a(p) = 1 for all p. - Wesley Ivan Hurt, May 21 2021

			a(28) = 4; (1,2), (1,7), (1,14), (2,7)
a(29) = 1; (1,29)
a(30) = 13; (1,2), (1,3), (1,5), (1,6), (1,10), (1,15), (1,30), (2,3), (2,5), (2,15), (3,5), (3,10), (5,6)
a(31) = 1; (1,31)

Cf. A008683, A074816.

with(numtheory): seq((3^nops(factorset(n))-1)/2, n=1..100); # Ridouane Oudra, Jan 27 2025

Table[Sum[Sum[MoebiusMu[i*k]^2 (1 - Ceiling[n/k] + Floor[n/k]) (1 - Ceiling[n/i] + Floor[n/i]), {i, k - 1}], {k, n}], {n, 100}]

Sum_{d1|n, d2|n, d1A008683).

a(n) = (A074816(n) - 1)/2. - Ridouane Oudra, Jan 27 2025

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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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A375230 The total number of infinitary divisors of the infinitary divisors of n.

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A340227 Number of pairs of divisors of n, (d1,d2), such that d1 < d2 and d1*d2 is squarefree.

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