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A051953 Cototient(n) := n - phi(n). 183
0, 1, 1, 2, 1, 4, 1, 4, 3, 6, 1, 8, 1, 8, 7, 8, 1, 12, 1, 12, 9, 12, 1, 16, 5, 14, 9, 16, 1, 22, 1, 16, 13, 18, 11, 24, 1, 20, 15, 24, 1, 30, 1, 24, 21, 24, 1, 32, 7, 30, 19, 28, 1, 36, 15, 32, 21, 30, 1, 44, 1, 32, 27, 32, 17, 46, 1, 36, 25, 46, 1, 48, 1, 38, 35, 40, 17, 54, 1, 48, 27 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,4

COMMENTS

Unlike totients, cototient(n+1) = cototient(n) never holds -- except 2-phi(2) = 3 - phi(3) = 1 -- because cototient(n) is congruent to n modulo 2. - Labos Elemer, Aug 08 2001

Theorem (L. Redei): b^a(n) == b^n (mod n) for every integer b. - Thomas Ordowski and Robert Israel, Mar 11 2016

LINKS

T. D. Noe, Table of n, a(n) for n = 1..10000

J. Browkin and A. Schinzel, On integers not of the form n-phi(n), Colloq. Math., 68 (1995), 55-58.

R. E. Jamison, The Helly bound for singular sums, Discrete Math., 249 (2002), 117-133.

P. Pollack, C. Pomerance, Some problems of Erdos on the sum-of-divisors function, For Richard Guy on his 99th birthday: May his sequence be unbounded, 2015, to appear.

Carl Pomerance and Hee-Sung Yang, Variant of a theorem of Erdos on the sum-of-proper-divisors function, Math. Comp., to appear (2014);

FORMULA

a(n) = n - A000010(n).

Equals Mobius transform (A054525) of A001065. - Gary W. Adamson, Jul 11 2008

a(A006881(n)) = sopf(A006881(n)) - 1; a(A000040(n)) = 1. - Wesley Ivan Hurt, May 18 2013

G.f.: sum(n>=1, A000010(n)*x^(2*n)/(1-x^n) ). - Mircea Merca, Feb 23 2014

From Ilya Gutkovskiy, Apr 13: (Start)

G.f.: -Sum_{k>=2} mu(k)*x^k/(1 - x^k)^2.

Dirichlet g.f.: zeta(s-1)*(1 - 1/zeta(s)). (End)

EXAMPLE

n = 12, phi(12) = 4 = |{1, 5, 7, 11}|, a(12) = 12 - phi(12) = 8, numbers not exceeding 12 and not coprime to 12: {2, 3, 4, 6, 8, 9, 10, 12}.

MAPLE

with(numtheory); A051953 := n->n-phi(n);

MATHEMATICA

Table[n - EulerPhi[n], {n, 1, 80}] (* Carl Najafi, Aug 16 2011 *)

PROG

(PARI) A051953(n) = n - eulerphi(n); \\ Michael B. Porter, Jan 28 2010

(Haskell)

a051953 n = n - a000010 n  -- Reinhard Zumkeller, Jan 21 2014

(Python)

from sympy.ntheory import totient

print [i - totient(i) for i in xrange(1, 101)] # Indranil Ghosh, Mar 17 2017

CROSSREFS

Cf. A000010, A005278, A001274, A098006, A054525, A001065.

Records: A065385, A065386.

Number of zeros in the n-th row of triangle A054521. - Omar E. Pol, May 13 2016

Sequence in context: A063717 A024994 A243329 * A079277 A066452 A007104

Adjacent sequences:  A051950 A051951 A051952 * A051954 A051955 A051956

KEYWORD

nonn,easy,nice,changed

AUTHOR

Labos Elemer, Dec 21 1999

STATUS

approved

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Last modified April 25 00:46 EDT 2017. Contains 285346 sequences.