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a(n) = sigma(d(d(d(n)))), where d(n) is the number of divisors of n.
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%I #13 Dec 08 2016 12:49:37

%S 1,3,3,3,3,3,3,3,3,3,3,4,3,3,3,3,3,4,3,4,3,3,3,4,3,3,3,4,3,4,3,4,3,3,

%T 3,3,3,3,3,4,3,4,3,4,4,3,3,4,3,4,3,4,3,4,3,4,3,3,3,7,3,3,4,3,3,4,3,4,

%U 3,4,3,7,3,3,4,4,3,4,3,4,3,3,3,7,3,3

%N a(n) = sigma(d(d(d(n)))), where d(n) is the number of divisors of n.

%C a(n) >= 3 for n > 1 with a(1) = 1. If n is a prime or a semiprime, a(n) = 3. The converse is not true since a(8) = 3, but 8 is neither a prime nor a semiprime.

%H Harvey P. Dale, <a href="/A243129/b243129.txt">Table of n, a(n) for n = 1..1000</a>

%F a(n) = A000203(A000005(A000005(A000005(n)))).

%e a(12) = 4; 12 has 6 divisors --> 6 has 4 divisors --> 4 has 3 divisors --> and the sum of the divisors of 3 is 4.

%p with(numtheory); A243129:=n->sigma(tau(tau(tau(n)))); seq(A243129(n), n=1..100);

%t Table[DivisorSigma[1, DivisorSigma[0, DivisorSigma[0, DivisorSigma[0, n]]]], {n, 100}]

%t Table[DivisorSigma[1,Nest[DivisorSigma[0,#]&,n,3]],{n,100}] (* _Harvey P. Dale_, Mar 24 2016 *)

%Y Cf. A000005, A000203.

%K nonn,easy

%O 1,2

%A _Wesley Ivan Hurt_, May 29 2014