A051027 a(n) = sigma(sigma(n)) = sum of the divisors of the sum of the divisors of n.

1, 4, 7, 8, 12, 28, 15, 24, 14, 39, 28, 56, 24, 60, 60, 32, 39, 56, 42, 96, 63, 91, 60, 168, 32, 96, 90, 120, 72, 195, 63, 104, 124, 120, 124, 112, 60, 168, 120, 234, 96, 252, 84, 224, 168, 195, 124, 224, 80, 128, 195, 171, 120, 360, 195, 360, 186, 234, 168, 480, 96

Offset: 1

Author

Judson Neer

Keywords

• easy
• nice
• nonn
• look

Examples

			a(2) = 4 because sigma(2)=1+2=3 and sigma(3)=1+3=4. - _Zak Seidov_, Aug 29 2012
		

References

• József Sándor, On the composition of some arithmetic functions, Studia Univ. Babeș-Bolyai, Vol. 34, No. 1 (1989), pp. 7-14.
• József Sándor and Borislav Crstici, Handbook of Number theory II, Kluwer Academic Publishers, 2004, Chapter 1, p. 39.

Links

• T. D. Noe, Table of n, a(n) for n=1..5000

Crossrefs

Cf. A000203.

Programs

Maple

with(numtheory): [seq(sigma(sigma(n)), n=1..100)];

Mathematica

DivisorSigma[1,DivisorSigma[1,Range[100]]] (* Zak Seidov, Aug 29 2012 *)

PARI

a(n)=sigma(sigma(n)); \\ Joerg Arndt, Feb 16 2014

Python

from sympy import divisor_sigma as sigma
def a(n): return sigma(sigma(n))
print([a(n) for n in range(1, 62)]) # Michael S. Branicky, Dec 05 2021

Formula

a(n) = A000203(A000203(n)). - Zak Seidov, Aug 29 2012

a(p) = sigma(p+1) = A000203(p+1), for p prime. - Wesley Ivan Hurt, Feb 14 2014

a(n) = 2*n iff n = 2^q with M_(q+1) = 2^(q+1) - 1 is a Mersenne prime, hence iff n = 2^q with q in A090748. - Bernard Schott, Aug 08 2019

a(n) >= 2*n for even n, with equality only when n = 2^k and 2^(k+1) - 1 is prime (Sándor, 1989). - Amiram Eldar, Mar 09 2021