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).

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

Author

Amarnath Murthy, Jun 13 2001

Keywords

• nonn

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

Examples

			sigma(d(12)) = sigma(6) = 12.
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Harry J. Smith)

Crossrefs

Cf. A000005, A000203, A001221, A008683, A062068.

Programs

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 *)

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

Formula

a(n) = A000203(A000005(n)). - Wesley Ivan Hurt, Sep 09 2013

Extensions

More terms from Jason Earls, Jun 19 2001