A119432 -id:A119432 - cp's OEIS frontend

A119433 Primitive elements of A119432.

2, 105, 165, 195, 3003, 3927, 4389, 4641, 4845, 5187, 5313, 5865, 6555, 7395, 7905, 8265, 8835, 9435, 10005, 10455, 10545, 10695, 10965, 11685, 11985, 12255, 12765, 13395, 13485, 13515, 14145, 14835, 15045, 15105, 15555, 16215, 16815, 17085

Offset: 1

Franklin T. Adams-Watters, May 19 2006

nonn

Elements of A119432 that are not divisible by any smaller element of that sequence.

Appears to be the lexicographically latest sequence of squarefree numbers such that all numbers with abundance >= -1 (see A103288) are divisible by one of the terms. - Peter Munn, Oct 19 2020

			From _Peter Munn_, Oct 23 2020: (Start)
Initial terms, showing factorization:
   n   a(n)
   1      2 = 2
   2    105 = 3 * 5 * 7
   3    165 = 3 * 5 * 11
   4    195 = 3 * 5 * 13
   5   3003 = 3 * 7 * 11 * 13
   6   3927 = 3 * 7 * 11 * 17
   7   4389 = 3 * 7 * 11 * 19
   8   4641 = 3 * 7 * 13 * 17
   9   4845 = 3 * 5 * 17 * 19
  10   5187 = 3 * 7 * 13 * 19
  11   5313 = 3 * 7 * 11 * 23
  12   5865 = 3 * 5 * 17 * 23
  13   6555 = 3 * 5 * 19 * 23
  14   7395 = 3 * 5 * 17 * 29
  15   7905 = 3 * 5 * 17 * 31
(End)

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Subsequence of A005117, A119432.

Cf. A103288.

Block[{a = {}}, Do[If[And[NoneTrue[a, Mod[i, #] == 0 &], 2 EulerPhi[i] <= i], AppendTo[a, i]], {i, 20000}]; a] (* Michael De Vlieger, Nov 05 2020 *)

2 followed by odd elements of A119431.

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

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

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

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A119433 Primitive elements of A119432.

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

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

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

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