cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-10 of 89 results. Next

A053408 Numbers k such that A003266(k) + 1 is prime.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 22, 28
Offset: 1

Views

Author

G. L. Honaker, Jr., Jan 08 2000

Keywords

Comments

Next term > 300. - Joerg Arndt, Aug 16 2014
The corresponding primes are given in A053413. - Joerg Arndt, Aug 17 2014
If it exists, a(11) > 1100. - Robert Price, May 27 2019

Links

Crossrefs

Cf. A000040, A000045, A003266, A053412, A053413, A059709.

Programs

  • Mathematica
    Select[Range[30], PrimeQ[Fibonorial[#] + 1] &] (* Robert Price, May 26 2019 *)
  • PARI
    ff(n)=prod(i=1, n, fibonacci(i));
for(n=1,10^6, if(ispseudoprime(ff(n)+1), print1(n,", "))); \\ Joerg Arndt, Aug 16 2014

Extensions

Definition edited by Daniel Forgues, Nov 29 2009
Edited definition, Joerg Arndt, Aug 17 2014

A123741 A second version of Fibonacci factorials besides A003266.

Original entry on oeis.org

1, 2, 24, 630, 52800, 11381760, 6738443712, 10487895163200, 43294107630090240, 469590163875486482400, 13388418681612808458240000, 1001088091286168023193223168000, 196239953628635168336022309340569600
Offset: 1

Views

Author

Wolfdieter Lang, Oct 13 2006

Keywords

Comments

The formula below is a generalization of n! = Product_{j=1..n} ((n+1) - j) with numbers k replaced by Fibonacci numbers F(k+1):=A000045(k+1), k>=1.
These numbers come up in Vandermonde determinants involving Fibonacci numbers [F(2),...,F(n+1)]. See A123742.

Examples

			n=3: (5-1)*(5-2)*(5-3) = 4*3*2 = 24;
n=4: (8-1)*(8-2)*(8-3)*(8-5) = 7*6*5*3 = 630.

Links

Crossrefs

Cf. A003266 (the usual Fibonacci factorials), A123742.

Programs

  • GAP
    F:=Fibonacci;; List([1..20], n-> Product([1..n], j-> F(n+2) - F(j+1))); # G. C. Greubel, Aug 10 2019
  • Magma
    F:=Fibonacci; [(&*[F(n+2)-F(j+1): j in [1..n]]): n in [1..20]] // G. C. Greubel, Aug 10 2019
  • Maple
    with(combinat): seq(mul(fibonacci(n+2)-fibonacci(j+1), j = 1..n), n = 1 .. 20); # G. C. Greubel, Aug 10 2019
  • Mathematica
    With[{F=Fibonacci}, Table[Product[F[n+2]-F[j+1],{j,n}], {n,20}]] (* G. C. Greubel, Aug 10 2019 *)
  • PARI
    vector(20, n, f=fibonacci; prod(j=1,n, f(n+2)-f(j+1))) \\ G. C. Greubel, Aug 10 2019
  • Sage
    f=fibonacci; [prod(f(n+2)-f(j+1) for j in (1..n)) for n in (1..20)] # G. C. Greubel, Aug 10 2019

Formula

a(n) = Product_{j=1..n} (F(n+2) - F(j+1)), n>=1.
a(n) ~ c * phi^(n*(n+2)) / 5^(n/2), where c = A276987 = QPochhammer(1/phi) and phi = A001622 = (1+sqrt(5))/2 is the golden ratio. - Vaclav Kotesovec, Mar 31 2021

A053413 Primes of the form A003266(n) + 1.

Original entry on oeis.org

2, 2, 3, 7, 31, 241, 3121, 65521, 1879127177606120717127879344567470740879360001, 1419564463863171507576408104556964008024375775796704645430601388670976000001
Offset: 1

Views

Author

G. L. Honaker, Jr., Jan 09 2000

Keywords

Comments

The corresponding n are given in A053408. - Joerg Arndt, Aug 16 2014

Links

Crossrefs

Cf. A059709, A053408.

Programs

  • Mathematica
    Select[Rest[FoldList[Times,1,Fibonacci[Range[80]]]+1],PrimeQ] (* Harvey P. Dale, Apr 25 2012 *)

Extensions

Edited by Daniel Forgues, Nov 30 2009
One more term (a(10)) from Harvey P. Dale, Apr 25 2012
Edited definition, Joerg Arndt, Aug 17 2014

A059709 Numbers k such that A003266(k) - 1 is prime.

Original entry on oeis.org

4, 5, 6, 7, 8, 14, 15
Offset: 1

Views

Author

Robert G. Wilson v, Feb 07 2001

Keywords

Comments

The a(n)-th almost-Fibonorial number is prime.
If it exists, a(8) > 1300. - Michael S. Branicky, Aug 17 2024

Links

Crossrefs

Cf. A003266, A053408.

Programs

  • Mathematica
    a = 1; Do[ a = a*Fibonacci[n]; If[ PrimeQ[a - 1], Print[n] ], {n, 1, 247} ]

Extensions

Definition corrected by Daniel Forgues, Nov 29 2009
Name edited by Michel Marcus, Jan 17 2024

A270653 Integers k such that A003266(k) is divisible by k.

Original entry on oeis.org

1, 5, 6, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80
Offset: 1

Views

Author

Altug Alkan, Mar 20 2016

Keywords

Comments

Note that this sequence is not the complement of A000057.
See A230359 for the prime terms of this sequence.

Examples

			11 is a term because 1*1*2*3*5*8*13*21*34*55*89 is divisible by 11.

Links

Crossrefs

Complement of A270777.
Cf. A000057, A003266, A230359.

Programs

  • Mathematica
    Select[Range@ 80, Divisible[Fibonorial@ #, #] &] (* Version 10, or *) Select[Range@ 80, Divisible[Product[Fibonacci@ k, {k, #}], #] &] (* Michael De Vlieger, Mar 23 2016 *)
  • PARI
    t(n) = prod(k=1, n, Mod(fibonacci(k), n));
for(n=1, 1e2, if(lift(t(n)) == 0, print1(n, ", ")));

A270777 Integers k such that A003266(k) is not divisible by k.

Original entry on oeis.org

2, 3, 4, 7, 23, 43, 67, 83, 103, 127, 163, 167, 223, 227, 283, 367, 383, 443, 463, 467, 487, 503, 523, 547, 587, 607, 643, 647, 683, 727, 787, 823, 827, 863, 883, 887, 907, 947, 983, 1063, 1123, 1163, 1187, 1283, 1303, 1327, 1367, 1423, 1447, 1487, 1543, 1567, 1583, 1607, 1627, 1663, 1667
Offset: 1

Views

Author

Altug Alkan, Mar 22 2016

Keywords

Examples

			4 is a term because 1*1*2*3 = 6 is not divisible by 4.

Links

Crossrefs

Complement of A270653.
Cf. A000057, A003266.

Programs

  • Mathematica
    Select[Range@ 1680, ! Divisible[Fibonorial@ #, #] &] (* Version 10, or *)
Select[Range@ 1680, ! Divisible[Product[Fibonacci@ k, {k, #}], #] &] (* Michael De Vlieger, Mar 27 2016 *)
  • PARI
    t(n) = prod(k=1, n, Mod(fibonacci(k), n));
for(n=1, 2000, if(lift(t(n)) != 0, print1(n, ", ")));

A359458 a(n) = A001911(n)*A003266(n+2).

Original entry on oeis.org

0, 2, 18, 180, 2640, 59280, 2096640, 118067040, 10659448800, 1548438091200, 362727075110400, 137200338475200000, 83862700757150515200, 82876486430812314240000, 132456397879190606981760000, 342431262483097194433458432000, 1432128704666605129972385934336000
Offset: 0

Views

Author

A.H.M. Smeets, Jan 03 2023

Keywords

Comments

Terms of the form 10^a(n)-1 for n>0 occur as large terms in the continued fraction expansion A359457 of the constant A359456.

Links

Crossrefs

Cf. A000045, A001911, A003266, A359456, A359457.

Formula

a(n) = (Sum_{i = 1..n} Fibonacci(i+1)) * (Product_{i = 1..n} Fibonacci(i+1)), with Fibonacci(k) = A000045(k).

A385608 a(n) = 2-adic valuation of A003266(n).

Original entry on oeis.org

0, 0, 0, 1, 1, 1, 4, 4, 4, 5, 5, 5, 9, 9, 9, 10, 10, 10, 13, 13, 13, 14, 14, 14, 19, 19, 19, 20, 20, 20, 23, 23, 23, 24, 24, 24, 28, 28, 28, 29, 29, 29, 32, 32, 32, 33, 33, 33, 39, 39, 39, 40, 40, 40, 43, 43, 43, 44, 44, 44, 48, 48, 48, 49, 49, 49, 52, 52, 52, 53, 53, 53
Offset: 0

Views

Author

Paolo Xausa, Jul 04 2025

Keywords

Links

Crossrefs

Cf. A000120, A003266, A007814, A385458, A385609.
Partial sums of A337923.

Programs

  • Mathematica
    A385608[n_] := 2*# + Quotient[n, 6] - DigitSum[#, 2] & [Quotient[n, 3]];
Array[A385608, 100, 0] (* or *)
Join[{0}, Accumulate[IntegerExponent[Fibonacci[Range[99]], 2]]]

Formula

a(n) = 2*floor(n/3) + floor(n/6) - A000120(floor(n/3)) (formula by David Radcliffe at A385458).
a(n) = A007814(A003266(n)).
For n >= 1, a(n) = Sum_{k=1..n} A337923(k).
a(3*k) = a(3*k+1) = a(3*k+2), for k >= 0.

A260622 a(n) is the sum of the positive divisors of A003266(n).

Original entry on oeis.org

1, 1, 3, 12, 72, 744, 10416, 270816, 9906624, 614210688, 55278961920, 8354817757440, 1955027355240960, 766650012876633600, 478623425047744204800, 492420437498707277414400, 786887859122934229308211200, 2148247421904894243053912064000
Offset: 1

Views

Author

Altug Alkan, Apr 30 2016

Keywords

Links

Crossrefs

Cf. A000203, A003266, A272122.

Programs

  • Mathematica
    a[n_] := DivisorSigma[1, Fibonorial[n]]; Array[a, 18] (* Amiram Eldar, Aug 09 2022 *)
  • PARI
    a(n) = sigma(prod(k=1, n, fibonacci(k)));

Formula

a(n) = A000203(A003266(n)).

A270532 Integers k such that A003266(p) is not divisible by p*(p+1)/2 where p is the k-th prime.

Original entry on oeis.org

1, 2, 4, 9, 14, 19, 23, 27, 31, 38, 39, 48, 49, 61, 73, 76, 86, 90, 91, 93, 96, 99, 101, 107, 111, 117, 118, 124, 129, 138, 143, 144, 150, 153, 154, 155, 161, 166, 179, 188, 192, 195, 208, 213, 217, 219, 224, 229, 236, 243, 247, 250, 253, 258, 261, 262, 269, 272, 276, 277, 283, 285, 292, 300
Offset: 1

Views

Author

Altug Alkan, Mar 18 2016

Keywords

Comments

See A000057 to corresponding prime numbers. In other words, this sequence is an indirect way to generate primes dividing all Fibonacci sequences.

Links

Crossrefs

Cf. A000057, A003266, A270475.

Programs

  • PARI
    f(n) = prod(k=1, n, fibonacci(k)); \\ A003266
isok(n) = (lift(Mod(f(prime(n)),(prime(n)*(prime(n)+1)/2))) != 0);
  • PARI
    isok(n) = my(p=prime(n), m=p*(p+1)/2); prod(k=1, p, Mod(fibonacci(k), m)); \\ Michel Marcus, May 14 2021
Showing 1-10 of 89 results. Next

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