

A007366


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


16



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
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OFFSET

1,2


COMMENTS

From Torlach Rush, Dec 21 2017: (Start)
Properties of the two inverses of the terms of this sequence, I1 and I2:
 I1 is nonsquarefree iff I2 is nonsquarefree.
 The sum of the Dirichlet inverse of Euler's totient function for I1 and I2 is always 0.
 The larger value of the Dirichlet inverse of Euler's totient function of I1 and I2 equals a(n) iff both I1 and I2 are squarefree.
 (I1+I2) is divisible by 3.
(End)
If the number of distinct prime factors of 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
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


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.
W. 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

T. D. Noe, Table of n, a(n) for n = 1..1000
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].
R. 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 *)


CROSSREFS

Cf. A000010, A007367.
Sequence in context: A303745 A303746 A303747 * A302280 A350627 A109958
Adjacent sequences: A007363 A007364 A007365 * A007367 A007368 A007369


KEYWORD

nonn


AUTHOR

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


STATUS

approved



