A309004 - OEIS

%I #36 Sep 27 2019 19:44:33

%S 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,

%T 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,

%U 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

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

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

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

%H Antti Karttunen, <a href="/A309004/b309004.txt">Table of n, a(n) for n = 1..75600</a>

%H <a href="/index/Eu#epf">Index entries for sequences computed from exponents in factorization of n</a>

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

%F a(A088860(k)) = k.

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

%F a(n) = A008480(A181819(n)). - _Antti Karttunen_, Sep 27 2019

%e 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}.

%e 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}.

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

%o (PARI)

%o A008480(n) = { my(es=factor(n)[, 2], s=vecsum(es)); s!/prod(i=1, #es, es[i]!); };

%o A181819(n) = factorback(apply(e->prime(e),(factor(n)[,2])));

%o A309004(n) = A008480(A181819(n)); \\ _Antti Karttunen_, Sep 27 2019

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

%K nonn

%O 1,12

%A _Amiram Eldar_, Jul 22 2019

%E More terms from _Antti Karttunen_, Sep 27 2019