A386760 - OEIS

A386760

Numbers k such that the number of decimal digits of the Lucas number L(k) is greater than the number of decimal digits of the Fibonacci number F(k).

5, 6, 10, 11, 15, 16, 20, 24, 25, 29, 30, 34, 35, 39, 44, 48, 49, 53, 54, 58, 59, 63, 67, 68, 72, 73, 77, 78, 82, 83, 87, 91, 92, 96, 97, 101, 102, 106, 111, 115, 116, 120, 121, 125, 126, 130, 134, 135, 139, 140, 144, 145, 149, 150, 154, 158, 159, 163, 164, 168

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OFFSET

1,1

COMMENTS

The difference in the number of decimal digits, A055642(L(k))-A055642(F(k)) = A060384(k)-A386758(k) is either zero or one. In fact, this difference is ceiling(beta-{k*alpha}), with alpha and beta as defined in the Formula section. This implies that, asymptotically, a fraction of beta=0.349485... of the Lucas numbers has one more decimal digit than the corresponding Fibonacci number. This gives the asymptotic behavior of the sequence as a(n)~n/beta. Conjecture: abs(a(n)-n/beta)<c, for some constant c.

LINKS

Hans J. H. Tuenter, Table of n, a(n) for n = 1..1000

FORMULA

The sequence consists of the integers k>=2, for which {k*alpha}<beta, where alpha=log_10(phi), beta=log_10(5)/2, {x}=x-floor(x), denotes the fractional part of x, log_10(phi) = A097348, and phi = (1+sqrt(5))/2 = A001622.

EXAMPLE

5 is a term since F(5)=5 has length 1 decimal digit, but L(5)=11 has length 2 decimal digits which is greater.

MATHEMATICA

Select[Range[168], IntegerLength[LucasL[#]]>IntegerLength[Fibonacci[#]]&] (* James C. McMahon, Aug 28 2025 *)

CROSSREFS

Cf. A000032, A000045, A001622, A055642, A060384, A097348, A386758.

Sequence in context: A374492 A064957 A008851 * A079259 A275018 A029772

Adjacent sequences: A386757 A386758 A386759 * A386761 A386762 A386763

KEYWORD

base,nonn,easy

AUTHOR

Hans J. H. Tuenter, Aug 13 2025

STATUS

approved