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A255133 Number of unitary divisors of odd Catalan numbers (A038003).

%I A255133 #16 Oct 04 2024 10:00:11
%S A255133 1,1,2,8,64,2048,1048576,68719476736,18446744073709551616,
%T A255133 166153499473114484112975882535043072,
%U A255133 822752278660603021077484591278675252491367932816789931674304512,4925250774549309901534880012517951725634967408808180833493536675530715221437151326426783281860614455100828498788352
%N A255133 Number of unitary divisors of odd Catalan numbers (A038003).
%H A255133 Chai Wah Wu, <a href="/A255133/b255133.txt">Table of n, a(n) for n = 0..14</a>
%F A255133 a(n) = A034444(A038003(n)).
%e A255133 A038003(4) = 9694845 which has 64 unitary divisors.
%t A255133 a[n_] := 2^PrimeNu[CatalanNumber[2^n-1]]; Array[a, 12, 0] (* _Amiram Eldar_, Oct 04 2024 *)
%o A255133 (Python)
%o A255133 from operator import mul
%o A255133 from functools import reduce
%o A255133 from sympy import factorint
%o A255133 A255133_list, c, s = [1, 1], {}, 3
%o A255133 for n in range(2, 2**16):
%o A255133     for p, e in factorint(4*n-2).items():
%o A255133         if p in c:
%o A255133             c[p] += e
%o A255133         else:
%o A255133             c[p] = e
%o A255133     for p, e in factorint(n+1).items():
%o A255133         if c[p] == e:
%o A255133             del c[p]
%o A255133         else:
%o A255133             c[p] -= e
%o A255133     if n == s:
%o A255133         c2 = 2**len(c)
%o A255133         A255133_list.append(c2)
%o A255133         s = 2*s+1
%o A255133 (PARI) a(n) = 1 << omega(binomial(2^(n+1)-2, 2^n-1)/(2^n)); \\ _Amiram Eldar_, Oct 04 2024
%Y A255133 Cf. A038003, A034444, A119861, A119908, A120274, A120275, A255132.
%K A255133 nonn
%O A255133 0,3
%A A255133 _Chai Wah Wu_, Feb 15 2015