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A062069
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).
24
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, 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, 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
OFFSET
1,2
COMMENTS
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]
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
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
LINKS
FORMULA
a(n) = A000203(A000005(n)). - Wesley Ivan Hurt, Sep 09 2013
EXAMPLE
sigma(d(12)) = sigma(6) = 12.
MAPLE
A062069:= (n-> numtheory[sigma](numtheory[tau](n))):
seq (A062069(n), n=1..40); # Jani Melik, Jan 25 2011
MATHEMATICA
Table[DivisorSigma[1, DivisorSigma[0, n]], {n, 1, 80}] (* Carl Najafi, Aug 16 2011 *)
PROG
(PARI) v=[]; for(n=1, 150, v=concat(v, sigma(numdiv(n)))); v
(PARI) { for (n=1, 1000, write("b062069.txt", n, " ", sigma(numdiv(n))) ) } \\ Harry J. Smith, Jul 31 2009
CROSSREFS
KEYWORD
nonn
AUTHOR
Amarnath Murthy, Jun 13 2001
EXTENSIONS
More terms from Jason Earls, Jun 19 2001
STATUS
approved