A118700 -id:A118700 - cp's OEIS frontend

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

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

A067800 Nonprime numbers k such that phi(k) > k/2.

Original entry on oeis.org

1, 9, 15, 21, 25, 27, 33, 35, 39, 45, 49, 51, 55, 57, 63, 65, 69, 75, 77, 81, 85, 87, 91, 93, 95, 99, 111, 115, 117, 119, 121, 123, 125, 129, 133, 135, 141, 143, 145, 147, 153, 155, 159, 161, 169, 171, 175, 177, 183, 185, 187, 189, 201, 203, 205, 207, 209, 213, 215

Offset: 1

Benoit Cloitre, Feb 07 2002

nonn

Sequence is similar to A014076(n) giving odd nonprimes. Only 3 terms = 105, 165, 195 are not in the sequence among 59 terms < 210.
Cototient(m) > totient(m) equivalent to 2*phi(m) < m; the missing terms mentioned here seem to form A036798. - Labos Elemer, May 08 2003
The number 9075 is not in this sequence, is in A014076 and is not in A036798, which means that the missing terms mentioned here do not form A036798 (cf. A118700). - R. J. Mathar, Aug 08 2007

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A000010 (phi), A014076, A036798, A051953, A083254, A118700.

Select[Range[250],!PrimeQ[#]&&EulerPhi[#]>#/2&] (* Harvey P. Dale, Aug 29 2021 *)

isok(k) = !isprime(k) && eulerphi(k) > k/2; \\ Amiram Eldar, May 08 2025

A119434 Odd n such that 2*phi(n) < n.

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

Franklin T. Adams-Watters, May 19 2006

nonn

Obviously 2*phi(n) = n is impossible for odd n. Odd elements of A054741 and A119432. This is not the same as A036798. 684411 = 3*7*13*23*109 is in this sequence but not in A036798. (This is may not be the smallest such value.) The primitive elements of this sequence are A119433, excluding the initial 2
If n is in the sequence, then so is every odd multiple of n. - Robert Israel, Jan 06 2017
The asymptotic density of this sequence is in the interval (0.01120, 0.01176) (Kobayashi, 2016). It is 1/2 less than the asymptotic density of A119432. The number of terms below 10^k for k = 3, 4, ... are 11, 109, 1152, 11076, 111927, 1124091, 11224403, 112074112, ... - Amiram Eldar, Oct 15 2020

Robert Israel, 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.

Cf. A000010, A036798, A054741, A118700, A119432, A119433, A299174.

select(t -> numtheory:-phi(t) < t/2, [seq(t,t=1..10000,2)]);

Select[Range[1, 10^4, 2], 2 EulerPhi[#] < #&] (* Jean-François Alcover, Apr 12 2019 *)

lista(nn) = forstep (n=1, nn, 2, if (n > 2*eulerphi(n), print1(n, ", "))) \\ Michel Marcus, Jul 04 2015

A036798 UNION A118700. - R. J. Mathar, Aug 08 2007

A119432 \ A299174. - Amiram Eldar, Oct 15 2020

A091495 Odd squarefree numbers k such that k/phi(k) > 2, where phi is Euler's totient function.

Original entry on oeis.org

105, 165, 195, 1155, 1365, 1785, 1995, 2145, 2415, 2805, 3003, 3045, 3135, 3255, 3315, 3705, 3795, 3885, 3927, 4305, 4389, 4485, 4515, 4641, 4785, 4845, 4935, 5115, 5187, 5313, 5565, 5655, 5865, 6045, 6105, 6195, 6405, 6555, 6765, 7035, 7095, 7215

Offset: 1

T. D. Noe, Jan 15 2004

nonn

Apparently the squarefree members of the sequence A036798. Note that 105, 165 and 195 are the only terms having 3 prime factors. Also note that all the numbers listed above have 3 as a factor. The smallest number of this form not divisible by 3 is 37182145 = 5*7*11*13*17*19*23.
From Amiram Eldar, Nov 21 2024: (Start)
If k is term and m is an odd squarefree number coprime to k, then k*m is also a term.
The numbers of terms that do not exceed 10^k, for k = 3, 4, ..., are 3, 58, 513, 5108, 52365, 523975, 5214831, 52103339, 521507571, ... . Apparently, the asymptotic density of this sequence exists and equals 0.00521... . (End)

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from T. D. Noe)

Cf. A000010, A036798, A056911, A118700.

lst={}; Do[f=FactorInteger[n]; s=Times@@Last/@f; If[s==1&&Times@@(1-1/(First/@f))<1/2, AppendTo[lst, n]], {n, 3, 10000, 2}]; lst
Select[Range[1,7301,2],SquareFreeQ[#]&&#/EulerPhi[#]>2&] (* Harvey P. Dale, Jul 10 2017 *)

is(k) = if(!(k%2), 0, my(f=factor(k)); issquarefree(f) && k / eulerphi(f) > 2); \\ Amiram Eldar, Nov 21 2024

cp's OEIS Frontend

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

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A067800 Nonprime numbers k such that phi(k) > k/2.

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Mathematica

PARI

A119434 Odd n such that 2*phi(n) < n.

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Maple

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PARI

Formula

A091495 Odd squarefree numbers k such that k/phi(k) > 2, where phi is Euler's totient function.

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PARI