|
%I #47 Dec 05 2023 08:11:42
%S 1,2,3,4,5,7,7,8,9,11,11,14,13,15,16,16,17,21,19,22,22,23,23,28,25,27,
%T 27,30,29,40,31,32,34,35,36,42,37,39,40,44,41,54,43,46,48,47,47,56,49,
%U 55,52,54,53,63,56,60,58
%N a(n) = (psi(n) + phi(n))/2.
%C This is (A001615 + A000010)/2. It is easy to see that this is always an integer.
%C If n is a power of a prime (including 1 and primes), then a(n) = n, and in any other case a(n) > n. - _M. F. Hasler_, Sep 09 2017
%C If n is in A006881, then a(n)=n+1. - _Robert Israel_, Feb 10 2019
%D Richard K. Guy, Unsolved Problems in Number Theory, 3rd Edition, Springer, 2004. See Section B41 (page 96 of 2nd ed., pages 147ff of 3rd ed.).
%H Hugo Pfoertner, <a href="/A291784/b291784.txt">Table of n, a(n) for n = 1..10000</a>
%H Marcin Mazur and Bogdan V. Petrenko, <a href="http://people.math.binghamton.edu/mazur/papers/pub5.pdf">Generalizations of Arnold's version of Euler's theorem for matrices</a>, Japanese Journal of Mathematics, 5:183-189, 2010.
%H N. J. A. Sloane, Three (No, 8) Lovely Problems from the OEIS, Experimental Mathematics Seminar, Rutgers University, Oct 05 2017, <a href="https://vimeo.com/237029685">Part I</a>, <a href="https://vimeo.com/237030304">Part 2</a>, <a href="https://oeis.org/A290447/a290447_slides.pdf">Slides</a>. (Mentions this sequence)
%F Sum_{k=1..n} a(k) = c * n^2 + O(n*log(n)), where c = 21/(4*Pi^2) = 0.531936... . - _Amiram Eldar_, Dec 05 2023
%p f:= proc(n) local P, p;
%p P:= numtheory:-factorset(n);
%p n*(mul((p-1)/p, p=P) + mul((p+1)/p, p=P))/2
%p end proc:
%p map(f, [$1..100]); # _Robert Israel_, Feb 10 2019
%t psi[n_] := If[n == 1, 1, n*Times @@ (1 + 1/FactorInteger[n][[All, 1]])];
%t a[n_] := (psi[n] + EulerPhi[n])/2;
%t Array[a, 100] (* _Jean-François Alcover_, Feb 25 2019 *)
%o (PARI) A291784(n)=(eulerphi(n)+n*sumdivmult(n,d,issquarefree(d)/d))\2 \\ _M. F. Hasler_, Sep 03 2017
%Y Cf. A000010, A001615, A006881, A291785, A291786, A291787, A291788.
%K nonn,easy
%O 1,2
%A _N. J. A. Sloane_, Sep 02 2017
|
