A247086 -id:A247086 - cp's OEIS frontend

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

Amiram Eldar, Nov 25 2021

nonn

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.

			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).

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

Cf. A000045, A010056, A000203, A001622, A017665, A017666.
Subsequences: A000396, A005820, A046060.
Similar sequences: A069070, A216780, A247086, A348658.

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]

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

A361469 a(n) = bigomega(A249670(A003961(n))).

Original entry on oeis.org

0, 3, 3, 3, 4, 4, 4, 7, 3, 7, 3, 4, 4, 5, 7, 6, 4, 6, 5, 7, 7, 6, 4, 6, 4, 5, 7, 5, 6, 8, 3, 9, 6, 7, 8, 6, 4, 6, 7, 11, 4, 8, 6, 4, 7, 5, 5, 7, 4, 5, 5, 3, 5, 8, 5, 9, 8, 9, 3, 8, 4, 6, 7, 7, 8, 7, 6, 7, 5, 9, 3, 8, 6, 5, 7, 6, 7, 8, 5, 10, 6, 7, 5, 6, 8, 7, 9, 10, 4, 10, 8, 5, 6, 6, 9, 10, 4, 7, 6, 5, 5, 6, 6, 7, 11

Offset: 1

Antti Karttunen, Mar 20 2023

nonn

Conjecture: There are no 1's in this sequence. If true, it would imply that there are no odd terms in A065997.

The first n with a(n) = 2 is 1684804. Note that A003961(1684804) = 5659641 is so far the only known odd term in A247086.

Antti Karttunen, Table of n, a(n) for n = 1..16384
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..100000
Index entries for sequences computed from indices in prime factorization
Index entries for sequences related to sigma(n)

Cf. A001222, A003961, A065997, A247086, A249670, A361468.

A003961(n) = { my(f=factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); }; \\ From A003961
A249670(n) = { my(ab = sigma(n)/n); numerator(ab)*denominator(ab); };
A361469(n) = bigomega(A249670(A003961(n)));

a(n) = A001222(A361468(n)) = A001222(A249670(A003961(n))).

cp's OEIS Frontend

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

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