

A062402


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


65



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
(list;
graph;
refs;
listen;
history;
text;
internal format)



OFFSET

1,3


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


REFERENCES

Krassimir T. Atanassov, One property of φ and σ functions, Bull. Number Theory Related Topics 13 (1989), pp. 2937.
A. Makowski and A. Schinzel, On the functions phi(n) and sigma(n), Colloq. Math. 13, 9599 (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, 18 (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, 3136 (2000).
F. Luca and C. Pomerance, On some problems of MakowskiSchinzel and Erdos concerning the arithmetical functions phi and sigma, Colloq. Math. 92, No. 1, 111130 (2002).


FORMULA

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


EXAMPLE

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


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 *)


PROG

(PARI) a(n)=sigma(eulerphi(n));
vector(150, n, a(n))
(Haskell)
a062402 = a000203 . a000010  Reinhard Zumkeller, Jan 04 2013
(Python)
from sympy import divisor_sigma, totient
print [divisor_sigma(totient(n)) for n in xrange(1, 101)] # Indranil Ghosh, Mar 18 2017


CROSSREFS

Cf. A000203, A000010, A062401, A096852, A096857, A096994, A096995, A033632.
Sequence in context: A083262 A122978 A119347 * A294015 A156838 A274845
Adjacent sequences: A062399 A062400 A062401 * A062403 A062404 A062405


KEYWORD

nonn,changed


AUTHOR

Jason Earls, Jul 08 2001


STATUS

approved



