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%I #36 Dec 15 2017 17:35:06
%S 1,3,3,4,3,7,3,7,4,7,3,12,3,7,7,6,3,12,3,12,7,7,3,15,4,7,7,12,3,15,3,
%T 12,7,7,7,13,3,7,7,15,3,15,3,12,12,7,3,18,4,12,7,12,3,15,7,15,7,7,3,
%U 28,3,7,12,8,7,15,3,12,7,15,3,28,3,7,12,12,7,15,3,18,6,7,3,28,7,7,7,15,3
%N a(n) = sigma(d(n)), where d(k) is the number of divisors function (A000005) and sigma(k) is the sum of divisors function (A000203).
%C a(1) = 1, a(p) = 3 for p = primes (A000040), a(pq) = 7 for pq = product of two distinct primes (A006881), a(pq...z) = 2^(k+1)-1 = A000225(k+1) for pq...z = product of k (k > 2) distinct primes p,q,...,z (A120944), a(p^k) = sigma(k+1) = A000203(k+1) for p^k = prime powers (A000961(n) for n > 1). Sequence {1,3,4,12} is finite sequence of numbers n such that sigma(tau(n)) = n. [_Jaroslav Krizek_, Jul 16 2009]
%C For semiprime n, a(n) is either 4 or 7. Also a(n) = d(n) + omega(n) + mu(n), the sum of three core sequences A000005, A001221 and A008683. When n is semiprime, a(n) is completely defined by the Mobius function as: a(n) = 4 + 3*mu(n). a(n) also has the fractal-like identities a(d(n)) = d(n) and a(n) = sigma(a(d(n))). - _Wesley Ivan Hurt_, Sep 02 2013
%C If n is a triprime (A014612), d(n) is 4, 6, or 8 and a(n) = sigma(d(n)) is 7, 12, or 15 respectively. Then a(n) = -d(n)^2/4 + 5*d(n) - 9. - _Wesley Ivan Hurt_, Sep 08 2013
%H Harry J. Smith, <a href="/A062069/b062069.txt">Table of n, a(n) for n=1,...,1000</a>
%F a(n) = A000203(A000005(n)). - _Wesley Ivan Hurt_, Sep 09 2013
%e sigma(d(12)) = sigma(6) = 12.
%p A062069:= (n-> numtheory[sigma](numtheory[tau](n))):
%p seq (A062069(n), n=1..40); # _Jani Melik_, Jan 25 2011
%t Table[DivisorSigma[1, DivisorSigma[0, n]], {n, 1, 80}] (* _Carl Najafi_, Aug 16 2011 *)
%o (PARI) v=[]; for(n=1,150,v=concat(v, sigma(numdiv(n)))); v
%o (PARI) { for (n=1, 1000, write("b062069.txt", n, " ", sigma(numdiv(n))) ) } \\ _Harry J. Smith_, Jul 31 2009
%Y Cf. A000005, A000203, A001221, A008683, A062068.
%K nonn
%O 1,2
%A _Amarnath Murthy_, Jun 13 2001
%E More terms from _Jason Earls_, Jun 19 2001
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