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Number of divisors of the n-th Bell number (A000110(n)).
2

%I #14 Jun 22 2022 14:49:33

%S 1,1,2,2,4,6,4,2,36,16,6,16,8,2,8,8,4,30,8,12,36,16,64,16,8,64,32,64,

%T 16,48,64,4,24,4,16,96,16,8,16,8,8,48,2,128,48,32,16,128,16,4,32,8,24,

%U 48,8,2,1728,8,8,32,8,128,8,128,16,24,64,8,24,16,16

%N Number of divisors of the n-th Bell number (A000110(n)).

%H Amiram Eldar, <a href="/A278973/b278973.txt">Table of n, a(n) for n = 0..104</a>

%F a(n) = tau(A000110(n)).

%F a(n) = A000005(A000110(n)).

%e Bell(17) = A000110(17) = 82864869804 = 2^2 * 3^4 * 255755771^1; exponents are 2, 4, 1, so its number of divisors is (2+1)*(4+1)*(1+1) = 3*5*2 = 30; thus a(17) = 30.

%e Bell(56) = A000110(56) = 6775685320645824322581483068371419745979053216268760300 = 2^2 * 3*2 * 5^2 * 7^1 * 43^1 * 481531^1 * 5134193^1 * 206802391^1 * 48920650786823172374961445939^1; exponents are 2, 2, 2, 1, 1, 1, 1, 1, 1, so its number of divisors is (2+1)^3 * (1+1)^6 = 1728; thus a(56) = 1728.

%t DivisorSigma[0,BellB[Range[0,70]]] (* _Harvey P. Dale_, Mar 04 2019 *)

%o (Python)

%o from sympy import bell, divisor_count

%o def A278973(n): return divisor_count(bell(n)) # _Chai Wah Wu_, Jun 22 2022

%Y Cf. A000005, A000110.

%K nonn

%O 0,3

%A _Jon E. Schoenfield_, Dec 02 2016