A062402 a(n) = sigma(phi(n)).

1, 1, 3, 3, 7, 3, 12, 7, 12, 7, 18, 7, 28, 12, 15, 15, 31, 12, 39, 15, 28, 18, 36, 15, 42, 28, 39, 28, 56, 15, 72, 31, 42, 31, 60, 28, 91, 39, 60, 31, 90, 28, 96, 42, 60, 36, 72, 31, 96, 42, 63, 60, 98, 39, 90, 60, 91, 56, 90, 31, 168, 72, 91, 63, 124, 42, 144, 63, 84, 60, 144

Offset: 1

Author

Jason Earls, Jul 08 2001

Keywords

• nonn

Comments

Makowski and Schinzel conjectured in 1964 that a(n) = sigma(phi(n)) >= n/2 for all n (B42 of Guy Unsolved Problems...). This has been verified for various classes of numbers and proved to be true in general if it is true for squarefree integers (see Cohen's paper). - Antonio G. Astudillo (afg_astudillo(AT)hotmail.com), Sep 07 2002

Atanassov proves the above conjecture. - Charles R Greathouse IV, Dec 06 2016

Examples

			a(9)= 12 because phi(9)= 6 and sigma(6)= 12.
		

References

• Krassimir T. Atanassov, One property of φ and σ functions, Bull. Number Theory Related Topics 13 (1989), pp. 29-37.
• A. Makowski and A. Schinzel, On the functions phi(n) and sigma(n), Colloq. Math. 13, 95-99 (1964).
• D. S. Mitrinovic et al., Handbook of Number Theory, Kluwer, p. 13.

Links

• T. D. Noe, Table of n, a(n) for n = 1..10000
• G. L. Cohen, On a conjecture of Makowski and Schinzel. Colloq. Math. 74, No. 1, 1-8 (1997).
• A. Grytczuk, F. Luca and M. Wojtowicz, On a conjecture of Makowski and Schinzel concerning the composition of the arithmetic functions sigma and phi, Colloq. Math. 86, No. 1, 31-36 (2000).
• F. Luca and C. Pomerance, On some problems of Makowski-Schinzel and Erdos concerning the arithmetical functions phi and sigma, Colloq. Math. 92, No. 1, 111-130 (2002).

Crossrefs

Cf. A000203, A000010, A062401, A096852, A096857, A096994, A096995, A033632.

Programs

Haskell

a062402 = a000203 . a000010  -- Reinhard Zumkeller, Jan 04 2013

Magma

[SumOfDivisors(EulerPhi(n)): n in [1..100]] //  Marius A. Burtea, Jan 19 2019

Maple

with(numtheory); A062402:=n->sigma(phi(n)); seq(A062402(k), k=1..100); # Wesley Ivan Hurt, Nov 01 2013

Mathematica

Table[DivisorSigma[1, EulerPhi[n]], {n, 1, 80}] (* Carl Najafi, Aug 16 2011 *)

PARI

a(n)=sigma(eulerphi(n));
vector(150,n,a(n))

Python

from sympy import divisor_sigma, totient
print([divisor_sigma(totient(n)) for n in range(1, 101)]) # Indranil Ghosh, Mar 18 2017

Formula

sigma(A062401(x)) = a(sigma(x)) or phi(a(x)) = A062401(phi(x)). - Labos Elemer, Jul 22 2004