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Revision History for A007366 (Bold, blue-underlined text is an addition; faded, red-underlined text is a deletion.)

Showing entries 1-10 | older changes
A007366
Numbers k such that phi(x) = k has exactly 2 solutions.
(history; published version)
#105 by Michael De Vlieger at Tue Nov 08 20:41:41 EST 2022
STATUS

proposed

approved

#104 by Jianing Song at Tue Nov 08 16:07:22 EST 2022
STATUS

editing

proposed

Discussion
Tue Nov 08
16:10
Jianing Song: "#({phi^-1(a(n))}) = 2. - Torlach Rush, Dec 22 2017": I don't feel like to keep. Of course I'm very likely to be wrong.
#103 by Jianing Song at Tue Nov 08 16:07:16 EST 2022
COMMENTS

The smaller of the solutions to phi(x) = a(n) is given by A271983(n). It is conjectured that the larger solution is 2*A271983(n); or equivalently, all terms in A271983 are odd. - Jianing Song, Nov 08 2022

CROSSREFS

Cf. A000010, A001221, A007367, A023900, A271983.

STATUS

approved

editing

#102 by Michael De Vlieger at Tue Nov 08 08:03:02 EST 2022
STATUS

reviewed

approved

#101 by Joerg Arndt at Tue Nov 08 05:54:19 EST 2022
STATUS

proposed

reviewed

#100 by Joerg Arndt at Tue Nov 08 05:53:58 EST 2022
STATUS

editing

proposed

#99 by Joerg Arndt at Tue Nov 08 05:53:46 EST 2022
COMMENTS

From Torlach Rush, Dec 21 2017: (Start)

Properties of the two solutions of the terms of this sequence, x_1 and x_2:

- x_1 is nonsquarefree iff x_2 is nonsquarefree.

- The sum of the Dirichlet inverse of Euler's totient function (A023900) for x_1 and x_2 is always 0.

- The larger value of A023900(x_1) and A023900(x_2) equals a(n) iff both x_1 and x_2 are squarefree.

- (x_1+x_2) is divisible by 3.

(End)

If the number of distinct prime factors of k (A001221(k)) equals the number of solutions of k = phi(x), then the greatest common divisor of the solutions is the least solution. - Torlach Rush, Jul 24 2018

For n > 1, a(n) is nonsquarefree if the lesser solution is divisible by an even number of primes. - Torlach Rush, Dec 22 2017

STATUS

proposed

editing

#98 by Torlach Rush at Tue Oct 11 20:07:30 EDT 2022
STATUS

editing

proposed

Discussion
Wed Oct 12
11:24
Joerg Arndt: Are these things proven? Otherwise everything is a conjecture.
Thu Oct 13
20:36
Torlach Rush: These are not conjectures. They follow from the fact that if only two solutions exist then the larger solution must be twice the smaller solution so, phi(x) = phi(2x) = phi(2)*phi(x) = 1*phi(x) as gcd(2, x) = 1.
22:35
Torlach Rush: Replace x_1 with x in the above,
Mon Nov 07
15:17
Jianing Song: No I cannot believe it. A271983 states explicitly that your statement "the larger solution must be twice the smaller solution" is a conjecture. And this seems like at the level of Carmichael's conjecture to me.
15:26
Jianing Song: By the way, you may be interested in the sequence A143510.
#97 by Torlach Rush at Tue Oct 11 20:05:23 EDT 2022
COMMENTS

Properties of the two inverses solutions of the terms of this sequence, I1 x_1 and I2x_2:

- I1 x_1 is nonsquarefree iff I2 x_2 is nonsquarefree.

- The sum of the Dirichlet inverse of Euler's totient function (A023900) for I1 x_1 and I2 x_2 is always 0.

- The larger value of A023900(I1x_1) and A023900(I2x_2) equals a(n) iff both I1 x_1 and I2 x_2 are squarefree.

- (I1x_1+I2x_2) is divisible by 3.

STATUS

proposed

editing

Discussion
Tue Oct 11
20:06
Torlach Rush: Changing inverses to solutions and In to x_n. Hope this helps.
#96 by Torlach Rush at Wed Oct 05 23:12:43 EDT 2022
STATUS

editing

proposed

Discussion
Tue Oct 11
05:49
Joerg Arndt: You never define what you mean by "inverse(s) of the terms of this sequence"!

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