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a(n) = sigma(phi(n)).
75

%I #47 Sep 08 2022 08:45:03

%S 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,

%T 39,28,56,15,72,31,42,31,60,28,91,39,60,31,90,28,96,42,60,36,72,31,96,

%U 42,63,60,98,39,90,60,91,56,90,31,168,72,91,63,124,42,144,63,84,60,144

%N a(n) = sigma(phi(n)).

%C 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

%C Atanassov proves the above conjecture. - _Charles R Greathouse IV_, Dec 06 2016

%D Krassimir T. Atanassov, One property of φ and σ functions, Bull. Number Theory Related Topics 13 (1989), pp. 29-37.

%D A. Makowski and A. Schinzel, On the functions phi(n) and sigma(n), Colloq. Math. 13, 95-99 (1964).

%D D. S. Mitrinovic et al., Handbook of Number Theory, Kluwer, p. 13.

%H T. D. Noe, <a href="/A062402/b062402.txt">Table of n, a(n) for n = 1..10000</a>

%H G. L. Cohen, <a href="http://matwbn.icm.edu.pl/ksiazki/cm/cm74/cm7411.pdf">On a conjecture of Makowski and Schinzel</a>. Colloq. Math. 74, No. 1, 1-8 (1997).

%H A. Grytczuk, F. Luca and M. Wojtowicz, <a href="http://matwbn.icm.edu.pl/ksiazki/cm/cm86/cm8615.pdf">On a conjecture of Makowski and Schinzel concerning the composition of the arithmetic functions sigma and phi</a>, Colloq. Math. 86, No. 1, 31-36 (2000).

%H F. Luca and C. Pomerance, <a href="http://dx.doi.org/10.4064/cm92-1-10">On some problems of Makowski-Schinzel and Erdos concerning the arithmetical functions phi and sigma</a>, Colloq. Math. 92, No. 1, 111-130 (2002).

%F sigma(A062401(x)) = a(sigma(x)) or phi(a(x)) = A062401(phi(x)). - _Labos Elemer_, Jul 22 2004

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

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

%t Table[DivisorSigma[1, EulerPhi[n]], {n, 1, 80}] (* _Carl Najafi_, Aug 16 2011 *)

%o (PARI) a(n)=sigma(eulerphi(n));

%o vector(150,n,a(n))

%o (Haskell)

%o a062402 = a000203 . a000010 -- _Reinhard Zumkeller_, Jan 04 2013

%o (Python)

%o from sympy import divisor_sigma, totient

%o print([divisor_sigma(totient(n)) for n in range(1, 101)]) # _Indranil Ghosh_, Mar 18 2017

%o (Magma) [SumOfDivisors(EulerPhi(n)): n in [1..100]] // _Marius A. Burtea_, Jan 19 2019

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

%K nonn

%O 1,3

%A _Jason Earls_, Jul 08 2001