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A065657 Numbers n divided by EulerPhi(n) in approximately the golden ratio, i.e., n minimizing |(k / EulerPhi(k)) - golden ratio phi| for numbers k with the same number of digits as n. 2
3, 9, 39, 117, 351, 507, 3417, 10251, 30753, 58089, 92259, 656121, 3870849, 98845053, 429262593, 7508684661 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

Since phi is irrational, n / EulerPhi(n) can only approximate phi. Probably an open question: can |(n / EulerPhi(n)) - phi| be made arbitrarily close to 0?

The listed terms have this property: for r = 1,...,5, all r-digit terms share the same set of prime factors. For example, all three 3-digit terms have prime factors 3 and 13. Furthermore, all listed terms are multiples of 3. I conjecture that these properties hold in general.

LINKS

Table of n, a(n) for n=1..16.

J. L. Pe, On Approximate Harmonic Division of n by phi(n)

EXAMPLE

|3 / EulerPhi(3) - phi| = .118034 (approximately) is minimal for all one-digit numbers, with 3/EulerPhi(3) = 9/EulerPhi(9) = 3/2.

|117 / EulerPhi(117) - phi| = .006966 (approximately) is minimal for all three-digit numbers, with 117/EulerPhi(117) = 351/Eulerphi(351) = 507/EulerPhi(507) = 13/8.

MAPLE

A065657 := proc(n) gr := (1+sqrt(5))/2 ; appr := 1000000+n ; dg := {} ;

        for k from 10^(n-1) to 10^n-1 do

                qual := abs(k/numtheory[phi](k)-gr) ;

                if dg = {} or is(qual < appr) then dg := {k} ; appr := qual ;

                elif qual = appr then dg := dg union {k} ;

                end if;

        end do:

        print(sort(dg)) ;

end proc:

for n from 1 do A065657(n) ; end do: # R. J. Mathar, Nov 16 2010

CROSSREFS

Cf. A001622, A000010.

Sequence in context: A020121 A270593 A059804 * A296102 A149026 A149027

Adjacent sequences:  A065654 A065655 A065656 * A065658 A065659 A065660

KEYWORD

nonn,base

AUTHOR

Joseph L. Pe, Dec 03 2001

EXTENSIONS

Edited and link fixed; would someone check this sequence? - Charles R Greathouse IV, Aug 02 2010

Checked up to and including the 5-digit terms, replaced prime factor example in the comment - R. J. Mathar, Nov 16 2010

a(12)-a(16) from Donovan Johnson, Sep 25 2011

STATUS

approved

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Last modified January 20 02:13 EST 2019. Contains 319320 sequences. (Running on oeis4.)