A083254 a(n) = 2*phi(n) - n.

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

Offset: 1

Author

Labos Elemer, May 08 2003

Keywords

• easy
• sign

Comments

Möbius transform of A033879, deficiency of n. - Antti Karttunen, Dec 26 2017

Examples

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

Links

• R. J. Mathar, Table of n, a(n) for n = 1..10000

Crossrefs

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

Programs

Maple

A083254 := proc(n)
    2*numtheory[phi](n)-n ;
end proc: # R. J. Mathar, Jan 13 2014

Mathematica

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

PARI

a(n)=2*eulerphi(n)-n \\ Charles R Greathouse IV, Feb 21 2013

Formula

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

From Antti Karttunen, Dec 26 2017: (Start)

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

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

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

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

(End)

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