A051953 Cototient(n) := n - phi(n).
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
Offset: 1
Examples
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}.
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
Crossrefs
Cf. A000010, A001065 (inverse Möbius transform), A005278, A001274, A083254, A098006, A049586, A051612, A053579, A054525, A062790 (Möbius transform), A063985 (partial sums), A063986, A290087.
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
Programs
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Haskell
a051953 n = n - a000010 n -- Reinhard Zumkeller, Jan 21 2014
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Maple
with(numtheory); A051953 := n->n-phi(n);
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Mathematica
Table[n - EulerPhi[n], {n, 1, 80}] (* Carl Najafi, Aug 16 2011 *) -
PARI
A051953(n) = n - eulerphi(n); \\ Michael B. Porter, Jan 28 2010
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Python
from sympy.ntheory import totient print([i - totient(i) for i in range(1, 101)]) # Indranil Ghosh, Mar 17 2017
Formula
a(n) = n - A000010(n).
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)
a(n) = Sum_{d|n, dA000010(d).
(End)
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