OFFSET

1,3

COMMENTS

In 1964, A. Mąkowski and A. Schinzel conjectured that sigma(phi(n))/n >= 1/2 for all n (see links Mąkowski & Schinzel and Graeme L. Cohen), equivalent to a(n) >= 0.

In 1992, K. Atanassov believed that he obtained a proof of this conjecture but his proof was valid only for certain special values of n (see link József Sándor).

Mrs. K. Kuhn checked that this inequality holds for all positive integers n having at most six prime factors (see A. Grytczuk, et al.).

REFERENCES

Richard K. Guy, Unsolved Problems in Number Theory, 3rd Edition, Springer, 2004, Section B42, pp. 150-152.

LINKS

Graeme L. Cohen, On a conjecture of Makowski and Schinzel, Colloquium Mathematicae, Vol. 74, No. 1 (1997), pp. 1-8.

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).

A. Mąkowski and A. Schinzel, On the functions phi(n) and sigma(n), Colloq. Math., Vol. 13, No. 1 (1964), pp. 95-99.

József Sándor, On the composition of some arithmetic functions, II, Journal of Inequalities in Pure and Applied Mathematics, 34 (1), May 2005.

FORMULA

EXAMPLE

phi(5) = 4, sigma(4) = 7 and a(5) = 2 * sigma(phi(5)) - 5 = 2*7-5 = 9.

phi(30) = 8, sigma(8) = 15 and a(30) = 2 * sigma(phi(30)) - 30 = 0.

MAPLE

with(numtheory):

E := seq(2*sigma(phi(n))-n, n=1..100);

MATHEMATICA

a[n_] := 2 * DivisorSigma[1, EulerPhi[n]] - n; Array[a, 100] (* Amiram Eldar, Jan 08 2021 *)

PROG

(PARI) a(n) = 2*sigma(eulerphi(n)) - n; \\ Michel Marcus, Jan 08 2021

CROSSREFS

KEYWORD

nonn

AUTHOR

Bernard Schott, Jan 08 2021

STATUS

approved