

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
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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 rdigit terms share the same set of prime factors. For example, all three 3digit 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 onedigit numbers, with 3/EulerPhi(3) = 9/EulerPhi(9) = 3/2.
117 / EulerPhi(117)  phi = .006966 (approximately) is minimal for all threedigit 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^(n1) to 10^n1 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 5digit 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



