A115405 -id:A115405 - cp's OEIS frontend

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

Labos Elemer, May 08 2003

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

			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.

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

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

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}]
```

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

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

A054741 Numbers m such that totient(m) < cototient(m).

Original entry on oeis.org

6, 10, 12, 14, 18, 20, 22, 24, 26, 28, 30, 34, 36, 38, 40, 42, 44, 46, 48, 50, 52, 54, 56, 58, 60, 62, 66, 68, 70, 72, 74, 76, 78, 80, 82, 84, 86, 88, 90, 92, 94, 96, 98, 100, 102, 104, 105, 106, 108, 110, 112, 114, 116, 118, 120, 122, 124, 126, 130, 132, 134, 136

Offset: 1

Labos Elemer, Apr 26 2000

nonn

For powers of 2, the two function values are equal.
Numbers m such that m/phi(m) > 2. - Charles R Greathouse IV, Sep 13 2013
Numbers m such that A173557(m)/A007947(m) < 1/2. - Antti Karttunen, Jan 05 2019
Numbers m such that there are powers of m that are abundant. This follows from abundancy and totient being multiplicative, with the abundancy for prime p of p^k being asymptotically p/(p-1) as k -> oo; given that p/(p-1) = p^k/phi(p^k) for k >= 1. - Peter Munn, Nov 24 2020

			For m = 20, phi(20) = 8, cototient(20) = 20 - phi(20) = 12, 8 = phi(20) < 20-phi(20) = 12; for m = 21, the opposite holds: phi = 12, 21-phi = 8.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
Mitsuo Kobayashi, A generalization of a series for the density of abundant numbers, International Journal of Number Theory, Vol. 12, No. 3 (2016), pp. 671-677.

A177712 is a subsequence. Complement: A115405.
Positions of negative terms in A083254.
Cf. A000010, A007947, A051953, A005408, A036798, A089684, A173557.
Cf. A323170 (characteristic function).
Complement of A000079\{1} within A119432.

Select[ Range[300], 2EulerPhi[ # ] < # &] (* Robert G. Wilson v, Jan 10 2004 *)

is(n)=n>2*eulerphi(n) \\ Charles R Greathouse IV, Sep 13 2013

m such that A000010(m) < A051953(m).

a(n) seems to be asymptotic to c*n with c=1.9566...... - Benoit Cloitre, Oct 20 2002 [It is an old theorem that a(n) ~ cn for some c, for any sequence of the form "m/phi(m) > k". - Charles R Greathouse IV, May 28 2015] [c is in the interval (1.9540, 1.9562) (Kobayashi, 2016). - Amiram Eldar, Feb 14 2021]

Erroneous comment removed by Antti Karttunen, Jan 05 2019

cp's OEIS Frontend

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

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