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A066678 Totients of the least numbers for which the totient is divisible by n. 8
1, 2, 6, 4, 10, 6, 28, 8, 18, 10, 22, 12, 52, 28, 30, 16, 102, 18, 190, 20, 42, 22, 46, 24, 100, 52, 54, 28, 58, 30, 310, 32, 66, 102, 70, 36, 148, 190, 78, 40, 82, 42, 172, 44, 180, 46, 282, 48, 196, 100, 102, 52, 106, 54, 110, 56, 228, 58, 708, 60, 366, 310, 126, 64 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

From Alonso del Arte, Feb 03 2017: (Start)

One of the less obvious consequences of Dirichlet's theorem on primes in arithmetic progression is that this sequence is well-defined for all positive integers.

Suppose n is a nontotient (see A007617). Obviously a(n) != n. Dirichlet's theorem assures us that, if nothing else, there are infinitely many primes of the form nk + 1 for k positive (and in this case, k > 1). Then phi(nk + 1) = nk, suggesting a(n) = nk corresponding to the smallest k.

Of course not all a(n) are 1 less than a prime, such as 8, 20, 24, 54, etc. (End)

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 1..5000

FORMULA

a(n) = A000010(A061026(n)).

EXAMPLE

a(23) = 46 because there is no solution to phi(x) = 23 but there are solutions to phi(x) = 46, like x = 47.

a(24) = 24 because there are solutions to phi(x) = 24, such as x = 35.

MATHEMATICA

EulerPhi[mulTotientList = ConstantArray[1, 70]; k = 1; While[Length[vac = Rest[Flatten[Position[mulTotientList, 1]]]] > 0, k++; mulTotientList[[Intersection[Divisors[EulerPhi[k]], vac]]] *= k]; mulTotientList] (* Vincenzo Librandi Feb 04 2017 *)

a[n_] := For[k=1, True, k++, If[Divisible[t = EulerPhi[k], n], Return[t]]];

Array[a, 64] (* Jean-Fran├žois Alcover, Jul 30 2018 *)

PROG

(Sage)

def A066678(n):

    s = 1

    while euler_phi(s) % n > 0: s += 1

    return euler_phi(s)

print [A066678(n) for n in (1..64)] # Peter Luschny, Feb 05 2017

CROSSREFS

Cf. A000010, A066674, A066675, A066676, A066677, A067005, A061026.

Sequence in context: A066043 A238642 A145019 * A306645 A113571 A119018

Adjacent sequences:  A066675 A066676 A066677 * A066679 A066680 A066681

KEYWORD

nonn

AUTHOR

Labos Elemer, Dec 22 2001

STATUS

approved

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Last modified February 25 22:55 EST 2020. Contains 332270 sequences. (Running on oeis4.)