A340461 a(n) = 2*sigma(phi(n)) - n.

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, 51, 28, 83, 0, 113, 30, 51, 28, 85, 20, 145, 40, 81, 22, 139, 14, 149, 40, 75, 26, 97, 14, 143, 34, 75, 68, 143, 24, 125, 64, 125, 54, 121, 2, 275, 82, 119, 62, 183, 18, 221, 58, 99

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

Table of n, a(n) for n=1..69.

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

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

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

a(n) = 0 for n in A074777.

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

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

Sequence in context: A257958 A329285 A211878 * A060481 A335228 A228492

Adjacent sequences: A340458 A340459 A340460 * A340462 A340463 A340464

Keyword

nonn

Author

Bernard Schott, Jan 08 2021

Status

approved