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a(n) = 2*sigma(phi(n)) - n.
0

%I #52 Jan 13 2021 22:21:14

%S 1,0,3,2,9,0,17,6,15,4,25,2,43,10,15,14,45,6,59,10,35,14,49,6,59,30,

%T 51,28,83,0,113,30,51,28,85,20,145,40,81,22,139,14,149,40,75,26,97,14,

%U 143,34,75,68,143,24,125,64,125,54,121,2,275,82,119,62,183,18,221,58,99

%N a(n) = 2*sigma(phi(n)) - n.

%C 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.

%C 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).

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

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

%H Graeme L. Cohen, <a href="https://eudml.org/doc/210499">On a conjecture of Makowski and Schinzel</a>, Colloquium Mathematicae, Vol. 74, No. 1 (1997), pp. 1-8.

%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 A. Mąkowski and A. Schinzel, <a href="https://eudml.org/doc/262922">On the functions phi(n) and sigma(n)</a>, Colloq. Math., Vol. 13, No. 1 (1964), pp. 95-99.

%H József Sándor, <a href="http://emis.impa.br/EMIS/journals/JIPAM/images/146_05_JIPAM/146_05_www.pdf">On the composition of some arithmetic functions, II</a>, Journal of Inequalities in Pure and Applied Mathematics, 34 (1), May 2005.

%F a(n) = 2*A062402(n) - n.

%F If n = A019434(k), then a(n) = 3*(A019434(k) - 2).

%F a(n) = 0 for n in A074777.

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

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

%p with(numtheory):

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

%t a[n_] := 2 * DivisorSigma[1, EulerPhi[n]] - n; Array[a, 100] (* _Amiram Eldar_, Jan 08 2021 *)

%o (PARI) a(n) = 2*sigma(eulerphi(n)) - n; \\ _Michel Marcus_, Jan 08 2021

%Y Cf. A000010, A000203, A019434 (Fermat primes), A062402, A074777 (a(n) = 0).

%K nonn

%O 1,3

%A _Bernard Schott_, Jan 08 2021