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A309004
The number of numbers with the same prime signature and set of distinct prime factors as n (including n).
5
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.
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
KEYWORD
nonn
AUTHOR
Amiram Eldar, Jul 22 2019
EXTENSIONS
More terms from Antti Karttunen, Sep 27 2019
STATUS
approved