A007366 Numbers k such that phi(x) = k has exactly 2 solutions. (Formerly M4685)

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

1,2

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

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.

Formula

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

Example

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

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 *)

Prog

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

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).

Sequence in context: A303745 A303746 A303747 * A302280 A350627 A367918

Adjacent sequences: A007363 A007364 A007365 * A007367 A007368 A007369

Keyword

nonn,changed

Author

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

Status

approved