login

Reminder: The OEIS is hiring a new managing editor, and the application deadline is January 26.

a(n) = number of divisors of the product of the divisors of n.
2

%I #14 Sep 08 2022 08:46:18

%S 1,2,2,4,2,9,2,7,4,9,2,28,2,9,9,11,2,28,2,28,9,9,2,65,4,9,7,28,2,125,

%T 2,16,9,9,9,100,2,9,9,65,2,125,2,28,28,9,2,126,4,28,9,28,2,65,9,65,9,

%U 9,2,637,2,9,28,22,9,125,2,28,9,125,2,247,2,9,28

%N a(n) = number of divisors of the product of the divisors of n.

%H Antti Karttunen, <a href="/A280684/b280684.txt">Table of n, a(n) for n = 1..10000</a>

%F a(n) = A000005(A007955(n)).

%F a(p) = 2 for p = primes (A000040).

%F a(n) = 4 for squares of primes (A001248).

%F a(n) = n for numbers 1, 2, 4, 28, 100, ...

%F a(n) = tau(n) for noncomposites (A008578).

%F a(n) = a(n+1) for numbers in A052213.

%e For n = 4; a(n) = tau (1*2*4) = tau(8) = 4.

%o (Magma) [#[d: d in Divisors(&*[d: d in Divisors(n)])]: n in [1..100]]

%o (PARI)

%o A007955(n) = if(issquare(n, &n), n^numdiv(n^2), n^(numdiv(n)/2)); \\ This function from _Charles R Greathouse IV_, Feb 11 2011

%o A280684(n) = numdiv(A007955(n)); \\ _Antti Karttunen_, May 19 2017

%Y Cf. A000005, A000040, A001248, A007955, A008578, A052213, A280685, A280582.

%K nonn

%O 1,2

%A _Jaroslav Krizek_, Jan 07 2017