A349687 Numbers whose numerator and denominator of their abundancy index are both Fibonacci numbers.

1, 2, 6, 15, 24, 26, 28, 84, 90, 96, 120, 270, 330, 496, 672, 1335, 1488, 1540, 1638, 8128, 24384, 27280, 44109, 68200, 131040, 447040, 523776, 18506880, 22256640, 33550336, 36197280, 38257095, 65688320, 91963648, 95472000, 100651008, 102136320, 176432256, 197308800

Offset

1,2

Comments

This sequence includes all the perfect numbers (A000396), 3-perfect numbers (A005820) and 5-perfect numbers (A046060).

The deficient terms, 1, 2, 15, 26, 1335, 44109, 38257095, ..., have an abundancy index which is a ratio of two consecutive Fibonacci numbers, 1/1, 3/2, 8/5, 21/13, 144/89, 610/377, 46368/28657, ..., which approaches the golden ratio phi = 1.618... (A001622) as the numerators and denominators get larger.

Links

Table of n, a(n) for n=1..39.

Michel Marcus, 85 terms (some terms might be missing in this list).

Example

2 is a term since sigma(2)/2 = 3/2 = Fibonacci(4)/Fibonacci(3).
15 is a term since sigma(15)/15 = 8/5 = Fibonacci(6)/Fibonacci(5).

Mathematica

fibQ[n_] := Or @@ IntegerQ /@ Sqrt[{5 n^2 - 4, 5 n^2 + 4}]; ai[n_] := DivisorSigma[1, n]/n; q[n_] := fibQ[Numerator[(ain = ai[n])]] && fibQ[Denominator[ain]]; Select[Range[10^6], q]

Prog

(PARI) isfib(n) = my(k=n^2); k+=(k+1)<<2; issquare(k) || (n>0 && issquare(k-8));
isok(n) = my(q=sigma(n)/n); isfib(numerator(q)) && isfib(denominator(q)); \\ Michel Marcus, Nov 25 2021

Crossrefs

Cf. A000045, A010056, A000203, A001622, A017665, A017666.

Subsequences: A000396, A005820, A046060.

Similar sequences: A069070, A216780, A247086, A348658.

Sequence in context: A276782 A227210 A177442 * A090979 A050508 A033298

Adjacent sequences: A349684 A349685 A349686 * A349688 A349689 A349690

Keyword

nonn

Author

Amiram Eldar, Nov 25 2021

Status

approved