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 A051953 Cototient(n) := n - phi(n). 291
 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. Paul Pollack and Carl Pomerance, Some problems of Erdős on the sum-of-divisors function, Trans. Amer. Math. Soc. Ser. B 3 (2016), 1-26. For Richard Guy on his 99th birthday. May his sequence be unbounded. Carl Pomerance and Hee-Sung Yang, Variant of a theorem of Erdős on the sum-of-proper-divisors function, Math. Comp. 83 (2014), 1903-1913. N. J. A. Sloane, Families of Essentially Identical Sequences, Mar 24 2021 (Includes this sequence) Eric Weisstein's World of Mathematics, Cototient 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 2017: (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) From Antti Karttunen, Sep 05 2018 & Apr 29 2022: (Start) Dirichlet convolution square of A317846/A046644 gives this sequence + A063524. a(n) = A003557(n) * A318305(n). a(n) = A000010(n) - A083254(n). a(n) = A318325(n) - A318326(n). a(n) = Sum_{d|n} A062790(d) = Sum_{d|n, dn-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 range(1, 101)]) # Indranil Ghosh, Mar 17 2017 CROSSREFS Cf. A000010, A001065, A005278, A001274, A083254, A098006, A049586, A051612, A053579, A054525, A062790 (Möbius transform), A063985 (partial sums), A063986, A290087. Records: A065385, A065386. Number of zeros in the n-th row of triangle A054521. - Omar E. Pol, May 13 2016 Cf. A063740 (number of k such that cototient(k) = n). - M. F. Hasler, Jan 11 2018 Sequence in context: A063717 A024994 A243329 * A079277 A066452 A007104 Adjacent sequences:  A051950 A051951 A051952 * A051954 A051955 A051956 KEYWORD nonn,easy,nice AUTHOR Labos Elemer, Dec 21 1999 STATUS approved

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Last modified November 26 03:08 EST 2022. Contains 358353 sequences. (Running on oeis4.)