A318325 -id:A318325 - cp's OEIS frontend

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

Labos Elemer, Dec 21 1999

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
Let S be the sum of the cototients of the divisors of n (A001065). S < n iff n is deficient, S = n iff n is perfect, and S > n iff n is abundant. - Ivan N. Ianakiev, Oct 06 2023

			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}.

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

Cf. A000010, A001065 (inverse Möbius transform), 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

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

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

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

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

from sympy.ntheory import totient
print([i - totient(i) for i in range(1, 101)]) # Indranil Ghosh, Mar 17 2017

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, dA007431(d)*(A000005(n/d)-1).
a(n) = A048675(A318834(n)) = A276085(A353564(n)). [These follow from the formula below]
a(n) = Sum_{d|n, dA000010(d).
a(n) = A051612(n) - A001065(n).
(End)

A318320 a(n) = (psi(n) - phi(n))/2.

Original entry on oeis.org

0, 1, 1, 2, 1, 5, 1, 4, 3, 7, 1, 10, 1, 9, 8, 8, 1, 15, 1, 14, 10, 13, 1, 20, 5, 15, 9, 18, 1, 32, 1, 16, 14, 19, 12, 30, 1, 21, 16, 28, 1, 42, 1, 26, 24, 25, 1, 40, 7, 35, 20, 30, 1, 45, 16, 36, 22, 31, 1, 64, 1, 33, 30, 32, 18, 62, 1, 38, 26, 60, 1, 60, 1, 39, 40, 42, 18, 72, 1, 56, 27, 43, 1, 84, 22, 45, 32, 52, 1, 96

Offset: 1

Antti Karttunen, Aug 26 2018

Antti Karttunen, Table of n, a(n) for n = 1..65537
Marcin Mazur and Bogdan V. Petrenko, Generalizations of Arnold's version of Euler's theorem for matrices, Japanese Journal of Mathematics, 5:183-189, 2010.

Cf. A000010, A001615, A291784, A292786, A318325, A318326, A318442.

Differs from A069359 for the first time at n=30, where a(30) = 32, while A069359(30) = 31.

psi[n_] := n * Times @@ (1 + 1/FactorInteger[n][[;; , 1]]); psi[1] = 1; a[n_] := (psi[n] - EulerPhi[n])/2; Array[a, 100] (* Amiram Eldar, Dec 05 2023 *)

A318320(n) = sumdiv(n,d,(-1==moebius(n/d))*d);

A318320(n) = ((n*sumdivmult(n, d, issquarefree(d)/d))-eulerphi(n))/2;

a(n) = (A001615(n) - A000010(n))/2 = A292786(n)/2.
a(n) = A291784(n) - A000010(n).
a(n) = A318326(n) + A318442(n).
Sum_{k=1..n} a(k) = c * n^2 + O(n*log(n)), where c = 9/(4*Pi^2) = 0.227972... . - Amiram Eldar, Dec 05 2023

A318326 a(n) = Sum_{d|n} [moebius(n/d) < 0]*(sigma(d)-d).

Original entry on oeis.org

0, 0, 0, 1, 0, 2, 0, 3, 1, 2, 0, 9, 0, 2, 2, 7, 0, 10, 0, 11, 2, 2, 0, 23, 1, 2, 4, 13, 0, 23, 0, 15, 2, 2, 2, 37, 0, 2, 2, 29, 0, 27, 0, 17, 13, 2, 0, 51, 1, 14, 2, 19, 0, 34, 2, 35, 2, 2, 0, 81, 0, 2, 15, 31, 2, 35, 0, 23, 2, 31, 0, 91, 0, 2, 15, 25, 2, 39, 0, 65, 13, 2, 0, 99, 2, 2, 2, 47, 0, 97, 2, 29, 2, 2, 2, 107, 0, 18, 19, 65, 0, 47

Offset: 1

Antti Karttunen, Aug 26 2018

nonn

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A001065, A051953, A291784, A318320, A318325, A318442.

A318326(n) = sumdiv(n,d,(-1==moebius(n/d))*(sigma(d)-d));

a(n) = Sum_{d|n} [A008683(n/d) == -1]*A001065(d).
a(n) = A318325(n) - A051953(n).
a(n) = A318320(n) - A318442(n).

A318442 a(n) = Sum_{d|n} [moebius(n/d) < 0]*A033879(d).

Original entry on oeis.org

0, 1, 1, 1, 1, 3, 1, 1, 2, 5, 1, 1, 1, 7, 6, 1, 1, 5, 1, 3, 8, 11, 1, -3, 4, 13, 5, 5, 1, 9, 1, 1, 12, 17, 10, -7, 1, 19, 14, -1, 1, 15, 1, 9, 11, 23, 1, -11, 6, 21, 18, 11, 1, 11, 14, 1, 20, 29, 1, -17, 1, 31, 15, 1, 16, 27, 1, 15, 24, 29, 1, -31, 1, 37, 25, 17, 16, 33, 1, -9, 14, 41, 1, -15, 20, 43, 30, 5, 1, -1, 18, 21, 32, 47, 22, -27, 1

Offset: 1

Antti Karttunen, Aug 26 2018

sign

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A033879, A083254, A318320, A318325, A318441.

A318442(n) = sumdiv(n,d,(-1==moebius(n/d))*(d+d-sigma(d)));

a(n) = Sum_{d|n} [A008683(n/d) == 1]*A033879(d).
a(n) = A318320(n) - A318326(n).
A083254(n) = A318441(n) - a(n).

A318441 a(n) = Sum_{d|n} [moebius(n/d) > 0]*A033879(d).

Original entry on oeis.org

1, 1, 2, 1, 4, 1, 6, 1, 5, 3, 10, -3, 12, 5, 7, 1, 16, -1, 18, -1, 11, 9, 22, -11, 19, 11, 14, 1, 28, -5, 30, 1, 19, 15, 23, -19, 36, 17, 23, -9, 40, -3, 42, 5, 14, 21, 46, -27, 41, 11, 31, 7, 52, -7, 39, -7, 35, 27, 58, -45, 60, 29, 24, 1, 47, 1, 66, 11, 43, 7, 70, -55, 72, 35, 30, 13, 59, 3, 78, -25, 41, 39, 82, -51, 63, 41, 55, -3, 88

Offset: 1

Antti Karttunen, Aug 26 2018

sign

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A033879, A083254, A291784, A318325, A318442.

A318441(n) = sumdiv(n,d,(1==moebius(n/d))*(d+d-sigma(d)));

a(n) = Sum_{d|n} [A008683(n/d) == 1]*A033879(d).
a(n) = A291784(n) - A318325(n).
A083254(n) = a(n) - A318442(n).

A318501 a(n) = SumXOR_{d|n} [moebius(n/d) > 0]*(sigma(d)-d).

Original entry on oeis.org

0, 1, 1, 3, 1, 6, 1, 7, 4, 8, 1, 17, 1, 10, 9, 15, 1, 20, 1, 23, 11, 14, 1, 39, 6, 16, 13, 29, 1, 43, 1, 31, 15, 20, 13, 49, 1, 22, 17, 49, 1, 55, 1, 41, 32, 26, 1, 75, 8, 42, 21, 47, 1, 70, 17, 67, 23, 32, 1, 97, 1, 34, 40, 63, 19, 79, 1, 59, 27, 75, 1, 107, 1, 40, 48, 65, 19, 91, 1, 109, 40, 44, 1, 131, 23, 46, 33, 95, 1, 155, 21

Offset: 1

Antti Karttunen, Aug 28 2018

nonn

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A003987, A008683, A318325, A318502, A318503.

A318501(n) = { my(v=0); fordiv(n, d, if(1==moebius(n/d), v=bitxor(v, sigma(d)-d))); (v); };

a(n) = A318502(n) XOR A318503(n).

cp's OEIS Frontend

A051953 Cototient(n) := n - phi(n).

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A318320 a(n) = (psi(n) - phi(n))/2.

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A318326 a(n) = Sum_{d|n} [moebius(n/d) < 0]*(sigma(d)-d).

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A318442 a(n) = Sum_{d|n} [moebius(n/d) < 0]*A033879(d).

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A318441 a(n) = Sum_{d|n} [moebius(n/d) > 0]*A033879(d).

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A318501 a(n) = SumXOR_{d|n} [moebius(n/d) > 0]*(sigma(d)-d).

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