A007366 Numbers k such that phi(x) = k has exactly 2 solutions.

1, 10, 22, 28, 30, 46, 52, 54, 58, 66, 70, 78, 82, 102, 106, 110, 126, 130, 136, 138, 148, 150, 166, 172, 178, 190, 196, 198, 210, 222, 226, 228, 238, 250, 262, 268, 270, 282, 292, 294, 306, 310, 316, 330, 342, 346, 358, 366, 372, 378, 382, 388, 418, 430, 438

Offset: 1

Author

N. J. A. Sloane, Mira Bernstein, Robert G. Wilson v

Keywords

• nonn

Comments

Contains {2*3^(6k+1): k >= 1} as a subsequence. This is the simplest proof for the infinity of these numbers (see Sierpiński, Exercise 12, p. 237). - Franz Vrabec, Aug 21 2021

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

Examples

			10 = phi(11) = phi(22).
		

References

• M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 840.
• Wacław Sierpiński, Elementary Theory of Numbers, Warszawa, 1964.
• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from T. D. Noe)
• M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
• Max Alekseyev, PARI/GP Scripts for Miscellaneous Math Problems (invphi.gp).
• Robert G. Wilson v, Letter to N. J. A. Sloane, Jul. 1992.

Crossrefs

Cf. A000010, A001221, A023900, A271983.

Number of solutions: A007617 (0), this sequence (2), A007367 (3), A060667 (4), A060668 (5), A060669 (6), A060670 (7), A060671 (8), A060672 (9), A060673 (10), A060674 (11), A060675 (12).

Programs

Maple

select(nops@numtheory:-invphi=2, [$1..1000]); # Robert Israel, Dec 20 2017

Mathematica

a = Table[ 0, {500} ]; Do[ p = EulerPhi[ n ]; If[ p < 501, a[ [ p ] ]++ ], {n, 1, 500} ]; Select[ Range[ 500 ], a[ [ # ] ] == 2 & ]
(* Second program: *)
With[{nn = 1325}, TakeWhile[Union@ Select[KeyValueMap[{#1, Length@ #2} &, PositionIndex@ Array[EulerPhi, nn]], Last@ # == 2 &][[All, 1]], # < nn/3 &] ] (* Michael De Vlieger, Dec 20 2017 *)

PARI

is(k) = invphiNum(k) == 2 \\ Amiram Eldar, Nov 16 2024, using Max Alekseyev's invphi.gp

Formula

#({phi^(-1)(a(n))}) = 2. - Torlach Rush, Dec 22 2017