A083254 - OEIS

%I #27 Sep 07 2023 13:51:24

%S 1,0,1,0,3,-2,5,0,3,-2,9,-4,11,-2,1,0,15,-6,17,-4,3,-2,21,-8,15,-2,9,

%T -4,27,-14,29,0,7,-2,13,-12,35,-2,9,-8,39,-18,41,-4,3,-2,45,-16,35,

%U -10,13,-4,51,-18,25,-8,15,-2,57,-28,59,-2,9,0,31,-26,65,-4,19,-22,69,-24,71,-2,5,-4,43,-30,77,-16,27,-2,81,-36,43,-2,25

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

%C Möbius transform of A033879, deficiency of n. - _Antti Karttunen_, Dec 26 2017

%H R. J. Mathar, <a href="/A083254/b083254.txt">Table of n, a(n) for n = 1..10000</a>

%F a(n) = totient(n) - cototient(n) = A000010(n) - A051953(n).

%F From _Antti Karttunen_, Dec 26 2017: (Start)

%F a(n) = A065620(A297153(n)) = A117966(A297154(n)).

%F a(n) = A297114(n) + A297115(n).

%F a(2n) = A297114(2n).

%F For all n >= 1, -a(A000010(n)) = A293516(n).

%F (End)

%F Sum_{k=1..n} a(k) ~ c * n^2, where  c = 6/Pi^2 - 1/2 = 0.107927... . - _Amiram Eldar_, Sep 07 2023

%e Case 1# - totient(x)-cototient[x] = 0 if x is a power of 2;

%e Case 2# - totient(x)>cototient[x] gives odd primes and also A067800, (= A014076 except probably A036798); e.g. n = 33: a(33) = 2.20-33 = 7; n = p prime: a(p) = p-2;

%e Case 3# - totient(x)<cototient[x] gives even numbers without powers of 2 and most probably A036798; e.g. n = 20: a(20) = -4; n = 105: a(105) = 2.48-105 = 96-105 = -9.

%p A083254 := proc(n)

%p     2*numtheory[phi](n)-n ;

%p end proc: # _R. J. Mathar_, Jan 13 2014

%t Table[2*EulerPhi[w]-w, {w, 1, 1000}]

%o (PARI) a(n)=2*eulerphi(n)-n \\ _Charles R Greathouse IV_, Feb 21 2013

%Y Cf. A000010, A051953, A000079, A014076, A033879, A067800, A036798, A083255, A115405, A293516, A297114, A297115, A297153, A297154.

%K easy,sign

%O 1,5

%A _Labos Elemer_, May 08 2003