A067800 -id:A067800 - 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

A036798 Odd numbers m such that there exists an even number k < m with phi(k) = phi(m).

Original entry on oeis.org

105, 165, 195, 315, 495, 525, 585, 735, 825, 945, 975, 1155, 1365, 1485, 1575, 1755, 1785, 1815, 1995, 2145, 2205, 2415, 2475, 2535, 2625, 2805, 2835, 2925, 3003, 3045, 3135, 3255, 3315, 3465, 3675, 3705, 3795, 3885, 3927, 4095, 4125, 4305, 4389, 4455

Offset: 1

David W. Wilson

nonn

These numbers m appear to satisfy cototient(m) > totient(m) or 2*phi(m) < m; they seem to be the missing terms mentioned in A067800. - Labos Elemer, May 08 2003
All elements in this sequence must have 2*phi(m) < m, but not the reverse. See A118700. - Franklin T. Adams-Watters, May 21 2006
The numbers of terms that do not exceed 10^k, for k = 3, 4, ..., are 11, 108, 1139, 11036, 111796, ... . Apparently, the asymptotic density of this sequence exists and equals 0.011... . - Amiram Eldar, Nov 21 2024

Robert Israel, Table of n, a(n) for n = 1..10000
Max Alekseyev, PARI/GP Scripts for Miscellaneous Math Problems (invphi.gp).

Cf. A000010, A067800, A083254.
Cf. A091495 (Odd, squarefree n such that n/phi(n) > 2).
Cf. A118700, A119434.

N:= 10^4: # to get all terms <= N
PhiE:= map(numtheory:-phi, [seq(i,i=2..N,2)]):
A:= NULL:
for n from 1 to N by 2 do
t:= numtheory:-phi(n);
if 2*t < n and member(t, PhiE[1..(n-1)/2]) then A:= A,n fi;
od:
A; # Robert Israel, Jan 06 2017

is(m) = m%2 && #select(k -> !(k%2) && k < m, invphi(eulerphi(m))) > 0; \\ Amiram Eldar, Nov 21 2024, using Max Alekseyev's invphi.gp

A083255 Odd composite numbers k such that cototient(k) - phi(k) = k - 2*phi(k) is an odd prime.

Original entry on oeis.org

165, 195, 5187, 5865, 7395, 10005, 15045, 16215, 21165, 22695, 27285, 37995, 42585, 44115, 50235, 57885, 59415, 60945, 64005, 310845, 346035, 347565, 486795, 635205, 707115, 890445, 979455, 994755, 1049835, 1070535, 1078815, 1083585, 1121745

Offset: 1

Labos Elemer, May 08 2003

nonn

Quite a number of terms are divisible by 3*5*17 = 255.

			m = 17425605 = 3*5*23*53*953 is a term since cototient(m) - phi(m) = 9712901 - 8712704 = 197 is an odd prime.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..369 from R. J. Mathar)

Cf. A000010, A051953, A036798, A067800, A083254.

Do[s=EulerPhi[n]; c=n-s; If[Greater[c, s]&&PrimeQ[c-s]&&OddQ[c-s]&&!PrimeQ[n], Print[{n, c-s, n/255}]], {n, 1, 10000000}]

A089684 Numbers k such that 2*phi(k) > k.

Original entry on oeis.org

1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, 33, 35, 37, 39, 41, 43, 45, 47, 49, 51, 53, 55, 57, 59, 61, 63, 65, 67, 69, 71, 73, 75, 77, 79, 81, 83, 85, 87, 89, 91, 93, 95, 97, 99, 101, 103, 107, 109, 111, 113, 115, 117, 119, 121, 123, 125, 127

Offset: 1

Benoit Cloitre, Jan 16 2004

Amiram Eldar, Table of n, a(n) for n = 1..10000
Amiram Eldar, Plot of a(n)/n for n = 2^(20..34).
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.
Vaclav Kotesovec, Plot of a(n)/n for n = 1..1000000.

Cf. A000010, A036798, A067800 (nonprime n such that phi(n) > n/2).
Cf. A036798, the missing odd numbers.
Complement of A119432.

lst={}; Do[If[2*EulerPhi[n]>n, AppendTo[lst, n]], {n, 200}]; lst (* T. D. Noe *)
Select[ Range[130], 2EulerPhi[ # ] > # &] (* Robert G. Wilson v, Jan 16 2004 *)

is(k) = 2*eulerphi(k) > k; \\ Amiram Eldar, Dec 01 2024

Asymptotic to c*n with c = 2.045...

2.04582 < c < 2.04818 (from the bounds on the asymptotic density of A119432 given by Kobayashi, 2016). - Amiram Eldar, Dec 01 2024

cp's OEIS Frontend

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

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A036798 Odd numbers m such that there exists an even number k < m with phi(k) = phi(m).

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A083255 Odd composite numbers k such that cototient(k) - phi(k) = k - 2*phi(k) is an odd prime.

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A089684 Numbers k such that 2*phi(k) > k.

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PARI

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