A245306 -id:A245306 - cp's OEIS frontend

A339173 Index of record values of A339082.

1, 3, 5, 9, 11, 15, 21, 27, 41, 45, 81, 129, 135, 357, 429, 431, 447, 507, 567, 569, 2969, 4721, 5385, 9309, 9675, 14429, 25559, 30755, 35997, 37509, 50831, 81837, 104909, 130019, 148089, 201105, 397377, 433779, 590039, 593687, 604709, 931515, 1049895, 1285605, 1636005, 1803057, 1968719, 2904351, 3244367, 3340365

Offset: 1

Chai Wah Wu, Nov 25 2020

nonn

Also index of record values of A335568.

Cf. A000040, A000045, A001605, A005478, A119984, A245306, A335568, A338762, A338794.

A339082(a(n)) = A119984(n).

For n > 1, a(n) = A001605(n+1)-2.

A339315 a(n) is the smallest number k such that k^2+1 divided by its largest prime factor is equal to F(2*n-1) for n > 0, or 0 if no such k exists, where F(n) is the Fibonacci sequence.

Original entry on oeis.org

1, 3, 8, 34, 55, 144, 610, 233, 12166, 2584, 4181, 68260, 46368, 75025, 3917414, 464656, 1346269, 16349962

Offset: 1

Michel Lagneau, Nov 30 2020

a(n) is the smallest number k such that A248516(k) = A001519(n) for n > 0, or 0 if no such k exists, where A001519(n) = F(2*n-1) (bisection of the Fibonacci sequence), with F(n) = A000045(n).
We observe that a(2 + 3m) = A001519(1 + 3m) = A000045(1 + 6m) for m = 2, 3, 4, 5. For n = 6, this property no longer works.
For k > 0, a(3k - 1) is odd, a(3k) and a(3k+1) are even.
We observe that a(n)^2 + 1 is the product of two prime Fibonacci numbers for n = 2, 3, 4, 6, 7.
The first 18 terms of the sequence are Fibonacci numbers, except a(9), a(12), a(15), a(16) and a(18).
The corresponding sequence b(n) = (a(n)^2+1)/ A001519(n) is 2, 5, 13, 89, 89, 233, 1597, 89, 92681, 1597, 1597, 162593, 28657, 28657, 29842993, 160373, 514229. We observe that a majority of terms of b(n) are prime Fibonacci numbers, except b(9), b(12), b(15) and b(16).

			a(4) = 34 because 34^2 + 1 = 13*89 = 1157, and 1157/89 = 13 = A248516(34) = A001519(4).
A curiosity: a(22) = 1134903170 = F(45) with F(45)^2 + 1 = F(43)*F(47) where F(43) and F(47) are prime Fibonacci numbers.

Cf. A000045, A001519, A001906, A002522, A005478, A014442, A245236, A245306, A248516, A281618.

with(numtheory):with(combinat,fibonacci):
nn:=100:n0:=20:
for n from 1 to n0 do:
  ii:=0:
  for m from 1 to 10^10 while(ii=0) do:
   x:=m^2+1:y:=factorset(x):n1:=nops(y):
   z:=x/y[n1]:
    if z = fibonacci(2*n-1)
     then
     ii:=1:printf(`%d %d \n`,n,m):
     else
    fi:
  od:
od:

a(n) = {my(k=1, f=fibonacci(2*n-1)); while ((k^2+1)/vecmax(factor(k^2+1)[,1]) != f, k++); k;} \\ Michel Marcus, Nov 30 2020

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A339173 Index of record values of A339082.

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A339315 a(n) is the smallest number k such that k^2+1 divided by its largest prime factor is equal to F(2*n-1) for n > 0, or 0 if no such k exists, where F(n) is the Fibonacci sequence.

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