A090206 -id:A090206 - cp's OEIS frontend

A090208 Partial sums of sequence (A090206) of consecutive nonprime Fibonacci numbers.

1, 2, 9, 29, 55, 89, 199, 521, 987, 1597, 3571, 6765, 10946, 17711, 28657, 64079, 121393, 196418, 317811, 514229, 1149851, 2178309, 3524578, 5702887, 9227465, 14930352, 24157817, 39088169, 63245986, 102334155

Offset: 0

Felix Tubiana, Jan 22 2004

Cf. A090206, A000045.

Total /@ Partition[ Select[Fibonacci /@ Range[0, 40], ! PrimeQ[#] &], 2, 1] (* Seth A. Troisi, May 14 2022 *)

F(0) = 0; F(1) = 1; F(n) = F(n-1) + F(n-2); F(0:9) = 0, 1, 1, 2, 3, 5, 8, 13, 21, 34 nonprimes: 0, 1, 1, 8, 21 sums of consecutive nonprimes: 1, 2, 9, 29

a(29) corrected by Seth A. Troisi, May 13 2022

A090228 Products of consecutive members of A090206 (nonprime Fibonacci numbers).

Original entry on oeis.org

0, 1, 8, 168, 714, 1870, 7920, 54288, 229970, 602070, 2550408, 10803704, 28284465, 74049690, 193864606, 821223648, 3478759200, 9107509825, 23843770274, 62423800998, 264431464440, 1120149658760, 2932589879121, 7677619978602

Offset: 0

Felix Tubiana, Jan 22 2004

Cf. A090206, A000045

F(0) = 0; F(1) = 1; F(n) = F(n-1) + F(n-2); F(0:9) = 0, 1, 1, 2, 3, 5, 8, 13, 21, 34 nonprimes: 0, 1, 1, 8, 21 product of consecutive nonprimes: 1, 8, 168

a(n) = A090206(n)*A090206(n+1). - David Wasserman, Oct 13 2005

More terms from David Wasserman, Oct 13 2005

A090229 Differences between consecutive nonprime Fibonacci numbers (A090206).

Original entry on oeis.org

1, 0, 7, 13, 13, 21, 89, 233, 233, 377, 1597, 1597, 2584, 4181, 6765, 28657, 28657, 46368, 75025, 121393, 514229, 514229, 832040, 1346269, 2178309, 3524578, 5702887, 9227465, 14930352

Offset: 0

Felix Tubiana, Jan 22 2004

Cf. A090206.

F(0) = 0; F(1) = 1; F(n) = F(n-1) + F(n-2); F(0:9) = 0, 1, 1, 2, 3, 5, 8, 13, 21, 34 nonprimes: 0, 1, 1, 8, 21 differences between consecutive nonprimes: 1, 0, 7, 13

A287273 Consider all ways of writing the composite Fibonacci number A090206(n+3) as product of two divisors d1d2 = d3d4 = ... The sequence a(n) gives the minimum sums of {d1+d2, d3+d4,...}.

Original entry on oeis.org

6, 10, 19, 16, 24, 42, 71, 68, 106, 150, 178, 447, 288, 431, 3026, 754, 1119, 1220, 1902, 2974, 3194, 19979, 5168, 142026, 7728, 13098, 13530, 136187, 20276, 62158, 34130, 52994, 119811, 92736, 138673, 6169970, 241310, 6379215, 392836, 55946694, 587943, 768686

Offset: 1

Michel Lagneau, May 22 2017

nonn

This sequence is included in A273227.

The primes of the sequence are 19, 71, 431, 19979,...

			a(5) = 24 because A090206(8) = 144 = 1*144 = 2*72 = 3*48 = 4*36 = 6*24 = 8*18 = 9*16 = 12*12 and the minimum sum of {145, 74, 51, 40, 30, 26, 25, 24} is 24.

Cf. A000045, A002808, A090206, A273227.

Function[n, Min@ Map[# + n/# &, Take[#, Ceiling[Length[#]/2]]] &@ Divisors@ n] /@ Select[Fibonacci@ Range@ 55, CompositeQ] (* Michael De Vlieger, May 24 2017 *)

for(n=1, 100, d=divisors(fibonacci(n)); t=#d; k=if(t%2, 2*d[t\2+1], d[t\2]+d[t\2+1]); print1(k", ")).

A049997 Numbers of the form Fibonacci(i)*Fibonacci(j).

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 8, 9, 10, 13, 15, 16, 21, 24, 25, 26, 34, 39, 40, 42, 55, 63, 64, 65, 68, 89, 102, 104, 105, 110, 144, 165, 168, 169, 170, 178, 233, 267, 272, 273, 275, 288, 377, 432, 440, 441, 442, 445, 466, 610, 699, 712, 714, 715, 720, 754

Offset: 0

Clark Kimberling

It follows from Atanassov et al. that a(n) << sqrt(phi)^n, which matches the trivial a(n) >> sqrt(phi)^n up to a constant factor. - Charles R Greathouse IV, Feb 06 2013

Conjecture: Fibonacci(m)*Fibonacci(n) with 2 < m < n is a perfect power only for (m,n) = (3,6). This has been verified for 2 < m < n <= 900. - Zhi-Wei Sun, Jan 02 2025

			25 is in the sequence since it is the product of two, not necessarily distinct, Fibonacci numbers, 5 and 5.
26 is in the sequence since it is the product of two Fibonacci numbers, 2 and 13.
27 is not in the sequence because there is no way whatsoever to represent it as the product of exactly two Fibonacci numbers.

Charles R Greathouse IV, Table of n, a(n) for n = 0..10000
K. T. Atanassov, Ron Knott, Kiyota Ozeki, A. G. Shannon, and László Szalay, Inequalities among related pairs of Fibonacci numbers, Fibonacci Quarterly 41:1 (2003), pp. 20-22.
Clark Kimberling, Orderings of products of Fibonacci numbers, Fibonacci Quarterly 42:1 (2004), pp. 28-35.
Zhi-Wei Sun, Perfect powers as products of two Fibonacci or Lucas numbersQuestion 485037 at MathOverflow, December 30, 2024.

Subsequence of A065108; apart from the first term, subsequence of A094563. Complement is A228523.
See A049998 for further information about this sequence. Cf. A080097.
Intersection with A059389 (sums of two Fibonacci numbers) is A226857.
Cf. also A090206, A005478.

Take[ Union@Flatten@Table[ Fibonacci[i]Fibonacci[j], {i, 0, 16}, {j, 0, i}], 61] (* Robert G. Wilson v, Dec 14 2005 *)

list(lim)=my(phi=(1+sqrt(5))/2, v=vector(log(lim*sqrt(5))\log(phi), i, fibonacci(i+1)), u=List([0]),t); for(i=1,#v,for(j=i,#v,t=v[i]*v[j];if(t>lim,break,listput(u,t)))); vecsort(Vec(u),,8) \\ Charles R Greathouse IV, Feb 05 2013

A182601 Position (index) of the smallest Fibonacci number in a sequence of at least n consecutive composite Fibonacci numbers.

Original entry on oeis.org

6, 8, 8, 18, 18, 30, 30, 30, 30, 30, 30, 30, 30, 48, 48, 48, 48, 48, 48, 48, 48, 48, 48, 48, 48, 48, 48, 48, 48, 48, 48, 48, 48, 48, 48, 84, 84, 84, 84, 84, 84, 84, 84, 84, 84, 84, 84, 138, 138, 138

Offset: 1

R. J. Mathar, Nov 22 2010

nonn

All entries are of the form A001605(.)+1.

The records in A182600 are 1, 3, 5, 13, 35, 47, 221,... with first differences 2, 2, 8, 22, 12, 174,... and these are the frequencies (repetitions) of the entries in this sequence here.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A005478, A090206.

# using b-file for A001605 as a list B001605
dB:= B001605[2..-1]-B001605[1..-2]:
f:= proc(n) local i;
 for i from 1 do if dB[i] >= n+1 then return B001605[i]+1 fi od
end proc:
map(f, [$1..100]); # Robert Israel, Dec 24 2024

min{ i: A000045(j) in A002808 for all j=i..i+n-1}.

A182602 Smallest Fibonacci number in a sequence of at least n consecutive composite Fibonacci numbers.

Original entry on oeis.org

8, 21, 21, 2584, 2584, 832040, 832040, 832040, 832040, 832040, 832040, 832040, 832040, 4807526976, 4807526976, 4807526976, 4807526976, 4807526976, 4807526976, 4807526976, 4807526976, 4807526976, 4807526976, 4807526976

Offset: 1

R. J. Mathar, Nov 22 2010

nonn

			a(2) = a(3) = 21 because (21, 34, 55) is a run of three consecutive Fibonacci composites, so 21 is in the sequence (21, 34) of at least 2 and in the sequence (21, 34, 55) of at least 3 consecutive composites in A000045.

Cf. A005478, A090206.

Fibonacci[#]&/@(Flatten[Table[SequencePosition[Table[If[CompositeQ[Fibonacci[n]],1,0],{n,100}],PadRight[ {},k,1],1],{k,100}],1][[;;,1]]) (* Harvey P. Dale, Aug 24 2024 *)

a(n) = A000045(A182601).

a(n) = min{ A000045(i): A000045(j) in A002808 for all j=i..i+n-1}.

A090207 Number of factors of n-th nonprime Fibonacci number.

Original entry on oeis.org

1, 1, 1, 3, 2, 2, 2, 6, 2, 3, 3, 5, 2, 4, 3, 2, 9, 3, 2, 4, 4, 7, 2, 4, 3, 2, 3, 10, 3, 3

Offset: 0

Felix Tubiana, Jan 22 2004

Cf. A090206, A000045.

F(0) = 0; F(1) = 1; F(n) = F(n-1) + F(n-2); F(0:9) = 0, 1, 1, 2, 3, 5, 8, 13, 21, 34 The nonprime numbers are: 0, 1, 1, 8, 21, 34 which have number of factors: 1, 1, 1, 3, 2, 2

A103736 Fibonacci numbers with nonprime indices.

Original entry on oeis.org

0, 1, 3, 8, 21, 34, 55, 144, 377, 610, 987, 2584, 6765, 10946, 17711, 46368, 75025, 121393, 196418, 317811, 832040, 2178309, 3524578, 5702887, 9227465, 14930352, 39088169, 63245986, 102334155, 267914296, 701408733, 1134903170, 1836311903

Offset: 0

Leroy Quet, Oct 01 2005

nonn

			55 is included because 55 is the 10th Fibonacci number and 10 is composite.

Cf. A000045, A018252, A090206.

with(combinat): a:=proc(n) if isprime(n)=false then fibonacci(n) else fi end: seq(a(n),n=0..52); # Emeric Deutsch, Oct 04 2005

Fibonacci /@ Select[Range[0, 46], ! PrimeQ[ # ] &] (* Robert G. Wilson v *)
Module[{nn=50,ix},ix=Table[If[PrimeQ[n],0,1],{n,0,nn}];Pick[Fibonacci[ Range[ 0,nn]],ix,1]] (* Harvey P. Dale, Apr 26 2018 *)

More terms from Robert G. Wilson v and Emeric Deutsch, Oct 04 2005

A358490 Composite Fibonacci numbers whose sum of prime factors (with multiplicity) is a prime.

Original entry on oeis.org

34, 75025, 196418, 701408733, 225851433717, 591286729879, 23416728348467685, 420196140727489673, 927372692193078999176, 16641027750620563662096, 114059301025943970552219, 1264937032042997393488322, 5358359254990966640871840, 2353412818241252672952597492098, 3807901929474025356630904134051

Offset: 1

Marc Kouyoumdjian, Nov 18 2022

nonn

			75025 is a term because it is a composite Fibonacci number whose sum of prime factors 5, 5 and 3001 is 3011, a prime number.

Shyam Sunder Gupta, Prime Curios! 34

Intersection of A000045 and A046363.
Intersection of A090206 and A100118.
Cf. A001414.

cp's OEIS Frontend

A090208 Partial sums of sequence (A090206) of consecutive nonprime Fibonacci numbers.

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A090228 Products of consecutive members of A090206 (nonprime Fibonacci numbers).

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A090229 Differences between consecutive nonprime Fibonacci numbers (A090206).

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A287273 Consider all ways of writing the composite Fibonacci number A090206(n+3) as product of two divisors d1*d2 = d3*d4 = ... The sequence a(n) gives the minimum sums of {d1+d2, d3+d4,...}.

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A049997 Numbers of the form Fibonacci(i)*Fibonacci(j).

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A182601 Position (index) of the smallest Fibonacci number in a sequence of at least n consecutive composite Fibonacci numbers.

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A182602 Smallest Fibonacci number in a sequence of at least n consecutive composite Fibonacci numbers.

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A090207 Number of factors of n-th nonprime Fibonacci number.

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A103736 Fibonacci numbers with nonprime indices.

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A358490 Composite Fibonacci numbers whose sum of prime factors (with multiplicity) is a prime.

A287273 Consider all ways of writing the composite Fibonacci number A090206(n+3) as product of two divisors d1d2 = d3d4 = ... The sequence a(n) gives the minimum sums of {d1+d2, d3+d4,...}.