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

Author

Amiram Eldar, Jul 22 2019

Keywords

• nonn

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.

Examples

			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}.
		

Links

• Antti Karttunen, Table of n, a(n) for n = 1..75600
• Index entries for sequences computed from exponents in factorization of n

Crossrefs

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

Programs

Mathematica

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

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

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

Extensions

More terms from Antti Karttunen, Sep 27 2019