cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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A071388 Numbers k such that the cardinality of the set of solutions to phi(x) = k is a prime.

Original entry on oeis.org

1, 2, 8, 10, 20, 22, 28, 30, 32, 44, 46, 48, 52, 54, 56, 58, 66, 70, 72, 78, 82, 92, 96, 102, 104, 106, 110, 116, 120, 126, 130, 132, 136, 138, 140, 148, 150, 156, 164, 166, 172, 178, 190, 196, 198, 204, 210, 212, 216, 220, 222, 226, 228, 238, 240, 250, 260, 262
Offset: 1

Views

Author

Labos Elemer, May 23 2002

Keywords

Comments

All terms except 1 are even. - Robert Israel, Mar 29 2020

Examples

			InvPhi(48) = {65,104,105,112,130,140,144,156,168,180,210} has 11 terms, so 48 is a term.

Links

Crossrefs

Cf. A000010, A007366, A007367, A014197, A058277, A060668, A060670, A060674, A063512, A071386-A071389.

Programs

  • Maple
    filter:= n -> isprime(nops(numtheory:-invphi(n))):
select(filter, [$1..400]); # Robert Israel, Mar 29 2020
  • PARI
    is(k) = isprime(invphiNum(k)); \\ Amiram Eldar, Nov 15 2024, using Max Alekseyev's invphi.gp

A293928 Totients phi(m) having one or more solutions m to phi(m)^2 = phi(phi(m)*m).

Original entry on oeis.org

1, 2, 4, 6, 8, 12, 16, 18, 20, 24, 32, 36, 40, 48, 54, 64, 72, 80, 84, 96, 100, 108, 120, 128, 144, 160, 162, 168, 192, 200, 216, 240, 252, 256, 272, 288, 312, 320, 324, 336, 360, 384, 400, 432, 440, 480, 486, 500, 504, 512, 544, 576, 588, 600, 624, 640, 648, 672, 684
Offset: 1

Views

Author

Torlach Rush, Oct 19 2017

Keywords

Comments

"Totients" are terms of A000010. - N. J. A. Sloane, Oct 22 2017
The smallest totient absent from the list is 10. This is because the totient inverses of 10, 11 and 22 are not solutions to phi(m)^2 = phi(phi(m)*m).
The formula is recursive. For example, taking a(22) we get the following: 11664 = phi(108*324), 1259712 = phi(11664*324), 136048896 = phi(1259712*324), ...
Where (if ever) does this first differ from A068997? - R. J. Mathar, Oct 30 2017
Apparently the set of the m is A151999. - R. J. Mathar, Mar 25 2024
If m satisfies phi(m)^2 = phi(phi(m)*m), then it satisfies phi(m)^(k+1) = phi(phi(m)^k*m) for all k >= 1. - Max Alekseyev, Dec 03 2024

Examples

			96 is a term since 96^2 = phi(96*288), with m=288 where phi(288) = 96.

Links

Crossrefs

Cf. A000010, A006511, A032447, A007366.
Subsequence of A002202.

Programs

  • PARI
    isok(n) = {my(iv = invphi(n)); if (#iv, for (m = 1, #iv, if (n^2 == eulerphi(n*iv[m]), return (1)););); return (0);} \\ using the invphi script by Max Alekseyev; Michel Marcus, Nov 01 2017

Extensions

More terms from Michel Marcus, Oct 24 2017
Definition simplified by Max Alekseyev, Dec 03 2024

A297475 Numbers n such that phi(x) = n for more than one value of x, and the smallest such x divides the largest.

Original entry on oeis.org

1, 2, 8, 10, 22, 28, 30, 44, 46, 52, 54, 56, 58, 66, 70, 78, 82, 92, 102, 104, 106, 110, 116, 126, 128, 130, 136, 138, 140, 148, 150, 164, 166, 172, 178, 190, 196, 198, 204, 210, 212, 222, 226, 228, 238, 250, 260, 262, 268, 270, 282, 292, 294, 296, 306, 310, 316, 330, 332, 342, 344, 346, 356, 358, 366, 368, 372
Offset: 1

Views

Author

Torlach Rush, Dec 30 2017

Keywords

Comments

The larger endpoint is always twice the value of the smaller endpoint.
Conjecture 1: The number of solutions, excluding endpoints is always 0, or an odd number. (known to n = 2 * 10^5)
Conjecture 2: If both endpoints are divisible by 5, then the number of solutions (excluding terms of A007366) is of the form 4k + 1. (known to n = 2 * 10^5)
A007366 is contained in this sequence and the number of solutions, excluding endpoints is always 0.
Terms of this sequence are totients with a single odd totient inverse.

Examples

			2 is in the sequence because {phi^-1(2)} = {3,4,6}, and 2 = 6 / 3.
8 is in the sequence because {phi^-1(8)} = {15,...,30}, and 2 = 30 / 15.
10 is in the sequence because {phi^-1(10)} = {11,22}, and 2 = 22 / 11.

Links

Crossrefs

Cf. A002181, A002202, A006511, A007366, A032447.

Programs

  • Mathematica
    With[{nn = 67}, Take[#, nn] &@ Keys@ Select[KeySort@ PositionIndex@ Array[EulerPhi, nn^2], IntegerQ[#2/#1] & @@ {First@ #, Last@ #} &]] (* Michael De Vlieger, Dec 31 2017 *)
  • PARI
    isok(n) = my(vx = invphi(n)); (#vx > 1) && ((vecmax(vx) % vecmin(vx)) == 0); \\ Michel Marcus, Jul 18 2018

Formula

2 = max({phi^-1(n)}) / min({phi^-1(n)}).
0 = A006511(n) mod A002181(n).

A271983 The smaller of a pair n, m such that phi(n) = phi(m) and there is no other k such that phi(n) = phi(k).

Original entry on oeis.org

1, 11, 23, 29, 31, 47, 53, 81, 59, 67, 71, 79, 83, 103, 107, 121, 127, 131, 137, 139, 149, 151, 167, 173, 179, 191, 197, 199, 211, 223, 227, 229, 239, 251, 263, 269, 271, 283, 293, 343, 307, 311, 317, 331, 361, 347, 359, 367, 373, 379, 383, 389, 419, 431, 439, 443, 463, 467, 479, 491, 499
Offset: 1

Views

Author

Geoffrey Critzer, Apr 17 2016

Keywords

Comments

If phi(x) = N has exactly two solutions, x = n and x = m, say (see A007366), it is conjectured that one of n and m is odd and the other even.
This sequence differs from A058340 in that it contains nonprime integers. The first few are 81, 121, 343, 361, 529, 649, 841, 961, 1219, 1331, 1537, 1633, ...

Examples

			81 is a term because phi(81) = phi(162) = 54 (= A007366(8)).

Crossrefs

Cf. A007366, A058340.

Programs

  • Mathematica
    (* takes about 2 minutes, can return the sequence up to terms less than 5760=Euler phi(13 primorial) *)
Prepend[Select[
   Table[Flatten[Position[Table[EulerPhi[n], {n, 1, 30030}], m]], {m,
     2, 500, 2}], Length[#] == 2 &][[All, 1]], 1]

Extensions

Edited by N. J. A. Sloane, Apr 22 2016 at the suggestion of Franklin T. Adams-Watters.

A296655 Numbers k such that phi(x) = k has a positive even number of solutions.

Original entry on oeis.org

1, 4, 6, 10, 12, 16, 18, 22, 24, 28, 30, 36, 42, 46, 52, 54, 58, 64, 66, 70, 78, 80, 82, 84, 88, 100, 102, 106, 110, 112, 126, 130, 136, 138, 148, 150, 160, 162, 166, 168, 172, 176, 178, 180, 184, 190, 196, 198, 200, 208, 210, 222, 224, 226, 228, 232, 238, 250
Offset: 1

Views

Author

Torlach Rush, Dec 17 2017

Keywords

Comments

When the number of solutions is 2, the sum of Sum_{d|x} d*mu(d) is always 0.
A007366 is contained in this sequence because it selects terms with the smallest even number of inverses.

Examples

			1 is a term because phi(1) has two inverses, 1, and 2.
6 is a term because phi(6) has four inverses, 7, 9, 14, and 18.
10 is a term because phi(10) has two inverses, 11, and 22.
18 is a term because phi(18) has four inverses, 19, 27, 38, 54.
348 is a term because phi(348) has six inverses, 349, 413, 531, 698, 826, and 1062.

Crossrefs

Cf. A007366, A014197, A032446, A071386.

Programs

  • Mathematica
    With[{nn = 1500}, TakeWhile[Union@ Select[KeyValueMap[{#1, Length@ #2} &, PositionIndex@ Array[EulerPhi, nn]], EvenQ@ Last@ # &][[All, 1]], # <= nn/6 &] ] (* Michael De Vlieger, Dec 20 2017 *)

Formula

0 = card({phi^-1(a(n))}) mod 2.

Extensions

Corrected and extended by Rémy Sigrist, Dec 19 2017
Previous Showing 11-15 of 15 results.

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