A309004 The number of numbers with the same prime signature and set of distinct prime factors as n (including n).

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

Offset

1,12

Comments

The number of permutations of the exponents in the prime signature of n.

The number of terms in the n-th row of A111470.

Links

Antti Karttunen, Table of n, a(n) for n = 1..75600

Index entries for sequences computed from exponents in factorization of n

Formula

a(n) = 1 if and only if n is a power of a squarefree number (A072774).

a(A088860(k)) = k.

a(A006939(k)) = A000142(k) = k!.

a(n) = A008480(A181819(n)). - Antti Karttunen, Sep 27 2019

Example

a(12) = a(18) = 2 since 12 = 2^2 * 3 and 18 = 3^2 * 2 have the same prime signature, (2, 1), and the same set of distinct prime factors, {2, 3}.
a(60) = a(90) = a(150) = 3 since 60 = 2^2 * 3 * 5, 90 = 3^2 * 2 * 5, and 150 = 5^2 * 2 * 3 have the same prime signature, (2, 1, 1), and the same set of distinct prime factors, {2, 3, 5}.

Mathematica

a[n_] := Multinomial @@ Tally[FactorInteger[n][[;; , 2]]][[;; , 2]]; Array[a, 100]

Prog

(PARI)
A008480(n) = { my(es=factor(n)[, 2], s=vecsum(es)); s!/prod(i=1, #es, es[i]!); };
A181819(n) = factorback(apply(e->prime(e), (factor(n)[, 2])));
A309004(n) = A008480(A181819(n)); \\ Antti Karttunen, Sep 27 2019

Crossrefs

Cf. A000142, A006939, A008480, A072774, A088860, A111470, A181819, A309308, A309309, A318762.

Sequence in context: A071625 A331592 A353745 * A355382 A304779 A361691

Adjacent sequences: A309001 A309002 A309003 * A309005 A309006 A309007

Keyword

nonn

Author

Amiram Eldar, Jul 22 2019

Extensions

More terms from Antti Karttunen, Sep 27 2019

Status

approved