A066659 -id:A066659 - cp's OEIS frontend

A015126 Least k such that phi(k) = phi(n).

1, 1, 3, 3, 5, 3, 7, 5, 7, 5, 11, 5, 13, 7, 15, 15, 17, 7, 19, 15, 13, 11, 23, 15, 25, 13, 19, 13, 29, 15, 31, 17, 25, 17, 35, 13, 37, 19, 35, 17, 41, 13, 43, 25, 35, 23, 47, 17, 43, 25, 51, 35, 53, 19, 41, 35, 37, 29, 59, 17, 61, 31, 37, 51, 65, 25, 67, 51, 69, 35, 71, 35, 73

Offset: 1

Vladeta Jovovic, Jan 12 2002

From Jianing Song, Nov 11 2022: (Start)
The first even term is a(33817088) = 16842752 (see A002181 and A143510).
Conjecture: a(n) is always odd for odd n. (End)

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

Cf. A000010, A028476, A066412, A066659.

Cf. A002181, A143510.

a(n) = {my(en = eulerphi(n)); k = 1; while (eulerphi(k) != en, k++); return (k);} \\ Michel Marcus, Jun 17 2013

a(n) = vecmin(select(x -> x<=n, invphi(eulerphi(n)))); \\ Amiram Eldar, Nov 14 2024, using Max Alekseyev's invphi.gp

A028476 Greatest k such that phi(k) = phi(n), where phi is Euler's totient function.

Original entry on oeis.org

2, 2, 6, 6, 12, 6, 18, 12, 18, 12, 22, 12, 42, 18, 30, 30, 60, 18, 54, 30, 42, 22, 46, 30, 66, 42, 54, 42, 58, 30, 62, 60, 66, 60, 90, 42, 126, 54, 90, 60, 150, 42, 98, 66, 90, 46, 94, 60, 98, 66, 120, 90, 106, 54, 150, 90, 126, 58, 118, 60, 198, 62, 126, 120, 210, 66, 134

Offset: 1

Vladeta Jovovic, Jan 12 2002

nonn

Every number in this sequence occurs at least twice. For all n > 6, a(n) > phi(n)^2 is impossible. - Alonso del Arte, Dec 31 2016

			phi(1) = 1 and phi(2) = 1 also. There is no greater k such that phi(k) = 1, so therefore a(1) = a(2) = 2.
phi(3) = phi(4) = phi(6) = 2, and there is no greater k such that phi(k) = 6, hence a(3) = a(4) = a(6) = 6.

Antti Karttunen, Table of n, a(n) for n = 1..10000 (computed from the b-file of A057826 provided by T. D. Noe)
Max Alekseyev, PARI/GP Scripts for Miscellaneous Math Problems (invphi.gp).

Cf. A000010, A015126, A057826, A066412, A066659, A071181.

Table[Module[{k = (2 Boole[n <= 6]) + #^2}, While[EulerPhi@ k != #, k--]; k] &@ EulerPhi@ n, {n, 120}] (* Michael De Vlieger, Dec 31 2016 *)

a(n) = invphiMax(eulerphi(n)); \\ Amiram Eldar, Nov 14 2024, using Max Alekseyev's invphi.gp

a(1) = a(2) = 2, for n > 2, a(n) = A057826(A000010(n)/2). - Antti Karttunen, Aug 07 2017

A043343 Numbers m such that there is no k > m such that phi(k) = phi(m), where phi is Euler's totient function.

Original entry on oeis.org

2, 6, 12, 18, 22, 30, 42, 46, 54, 58, 60, 62, 66, 90, 94, 98, 106, 118, 120, 126, 134, 138, 142, 150, 158, 162, 166, 174, 198, 206, 210, 214, 240, 242, 250, 254, 262, 270, 274, 276, 278, 282, 294, 298, 302, 318, 330, 334, 346, 348, 354, 358, 378, 382, 394, 398

Offset: 1

Vladeta Jovovic, Jan 12 2002

nonn

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..980 from Paul Tek)
Max Alekseyev, PARI/GP Scripts for Miscellaneous Math Problems (invphi.gp).

Cf. A000010, A028476, A066659.

Flatten@ Position[#, 0] &@ Table[k = n + 1; While[And[k <= 2 n + 1, EulerPhi@ k != EulerPhi@ n], k++]; Boole[k < 2 n + 1] k, {n, 400}] (* Michael De Vlieger, Dec 31 2016 *)

is(k) = #select(x -> x>k, invphi(eulerphi(k))) == 0; \\ Amiram Eldar, Nov 12 2024, using Max Alekseyev's invphi.gp

A066659(a(n)) = 0.

A028476(a(n)) = a(n).

Offset corrected by Paul Tek, Sep 25 2015

Definition clarified by Alonso del Arte, Dec 31 2016

A262599 Lexicographically earliest sequence of distinct terms such that, for any n > 0, phi(a(n)) = phi(n) (where phi denotes the Euler totient function), and a(n) > n if possible.

Original entry on oeis.org

2, 1, 4, 6, 8, 3, 9, 10, 14, 12, 22, 5, 21, 18, 16, 20, 32, 7, 27, 24, 26, 11, 46, 30, 33, 28, 38, 36, 58, 15, 62, 34, 44, 40, 39, 42, 57, 54, 45, 48, 55, 13, 49, 50, 52, 23, 94, 60, 86, 66, 64, 56, 106, 19, 75, 70, 63, 29, 118, 17, 77, 31, 74, 68, 104, 25

Offset: 1

Paul Tek, Sep 25 2015

nonn

This is a permutation of the positive integers, with inverse A262603.
If the Carmichael's totient function conjecture is true, then this sequence has no fixed point.
For any n > 0, the orbit of n is finite, with length A066412(n).

			phi(n) = 6 iff n is in { 7, 9, 14, 18 }.
Hence: a(7) = 9, a(9) = 14, a(14) = 18, a(18) = 7.

Paul Tek, Table of n, a(n) for n = 1..10000
Paul Tek, C program for this sequence
Index entries for sequences that are permutations of the natural numbers

Cf. A049283, A066412, A066659, A262603 (inverse).

C
```
// See Links section for C program.
```

a(n) = max(A066659(n), A049283(A000010(n))), for any n > 0.

A272328 Number of integers 1<=k<=n such that phi(n)=phi(n+k) where phi is Euler's totient function A000010.

Original entry on oeis.org

1, 0, 2, 1, 2, 0, 2, 2, 2, 1, 1, 0, 2, 1, 4, 3, 2, 0, 2, 2, 4, 0, 1, 1, 3, 3, 2, 2, 1, 0, 1, 4, 3, 3, 5, 1, 3, 1, 6, 2, 3, 0, 2, 2, 7, 0, 1, 1, 2, 1, 5, 6, 1, 0, 5, 5, 5, 0, 1, 0, 4, 0, 5, 5, 4, 0, 1, 4, 2, 4, 1, 3, 6, 4, 6, 3, 5, 2, 1, 3, 1, 5, 1, 1, 4, 1, 2

Offset: 1

Tom Edgar, Apr 25 2016

nonn

If n is odd, then phi(n) = phi(2n) so that a(n)>=1.
If n is a member of A043343, then a(n)=0.
It seems that every nonnegative integer appears in this sequence.

			For n=2: phi(2) = 1; whereas phi(2+1) = 2 and phi(2+2) = 2. Thus a(2) = 0.
For n=5: phi(5) = 4, phi(5+1)=2, phi(5+2)=6, phi(5+3) = 4, phi(5+4) = 6, and phi(5+5) = 4. Since phi(5) = phi(5+3) = phi(5+5), a(5) = 2.

Chai Wah Wu, Table of n, a(n) for n = 1..10000

Cf. A043343, A066659, A081373.

Table[Count[Range@ n, k_ /; EulerPhi@ n == EulerPhi[n + k]], {n, 120}] (* Michael De Vlieger, Apr 25 2016 *)

a(n) = my(x=eulerphi(n)); sum(k=1, n, eulerphi(n+k) == x); \\ Michel Marcus, Mar 08 2020

from sympy import totient
nmax = 10**4
philist = [totient(i) for i in range(1,2*nmax+1)]
A272328_list = [philist[i+1:2*(i+1)].count(philist[i]) for i in range(nmax)] # Chai Wah Wu, Apr 26 2016

[sum([1 for k in [1..n] if euler_phi(n)==euler_phi(n+k)]) for n in [1..1000]]

A066705 Greatest k < n such that phi(k) = phi(n) if such k exists, otherwise 0.

Original entry on oeis.org

0, 1, 0, 3, 0, 4, 0, 5, 7, 8, 0, 10, 0, 9, 0, 15, 0, 14, 0, 16, 13, 11, 0, 20, 0, 21, 19, 26, 0, 24, 0, 17, 25, 32, 0, 28, 0, 27, 35, 34, 0, 36, 0, 33, 39, 23, 0, 40, 43, 44, 0, 45, 0, 38, 41, 52, 37, 29, 0, 48, 0, 31, 57, 51, 0, 50, 0, 64, 0, 56, 0, 70, 0, 63, 55, 74, 61, 72, 0, 68, 0

Offset: 1

Vladeta Jovovic, Jan 14 2002

nonn

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

Cf. A000010, A015126, A028476, A043343, A066659.

a(n) = {my(v = select(x -> xAmiram Eldar, Nov 14 2024, using Max Alekseyev's invphi.gp

cp's OEIS Frontend

A015126 Least k such that phi(k) = phi(n).

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A028476 Greatest k such that phi(k) = phi(n), where phi is Euler's totient function.

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A043343 Numbers m such that there is no k > m such that phi(k) = phi(m), where phi is Euler's totient function.

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A262599 Lexicographically earliest sequence of distinct terms such that, for any n > 0, phi(a(n)) = phi(n) (where phi denotes the Euler totient function), and a(n) > n if possible.

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A272328 Number of integers 1<=k<=n such that phi(n)=phi(n+k) where phi is Euler's totient function A000010.

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A066705 Greatest k < n such that phi(k) = phi(n) if such k exists, otherwise 0.

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