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.

3, 9, 39, 117, 351, 507, 3417, 10251, 30753, 58089, 92259, 656121, 3870849, 98845053, 429262593, 7508684661

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| = 0.118034 (approximately) is minimal for all one-digit numbers, with 3/EulerPhi(3) = 9/EulerPhi(9) = 3/2.
|117/EulerPhi(117) - phi| = 0.006966 (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

Mathematica

run[k_] := run[k] = SplitBy[Sort[Table[{Abs[n/EulerPhi[n] - GoldenRatio] // N, n}, {n, 10^(k-1), 10^k-1}]], First][[1]][[All, 2]];
Table[Print[run[k]]; run[k], {k, 1, 7}] // Flatten (* Jean-François Alcover, Jun 22 2019 *)

Crossrefs

Cf. A001622, A000010.

Sequence in context: A020121 A270593 A059804 * A296102 A149026 A149027

Adjacent sequences: A065654 A065655 A065656 * A065658 A065659 A065660

Keyword

nonn,base,more

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