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.

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A082602 Number of primes between successive Fibonacci numbers (including possibly the Fibonacci numbers themselves).

Original entry on oeis.org

0, 1, 2, 2, 2, 2, 3, 3, 5, 8, 11, 17, 24, 37, 55, 85, 126, 198, 297, 458, 704, 1088, 1674, 2602, 4029, 6263, 9738, 15187, 23705, 36981, 57909, 90550, 142033, 222855, 349862, 549903, 865019, 1361581, 2145191, 3381318, 5334509, 8419528, 13298631, 21014892
Offset: 1

Views

Author

Hauke Worpel (hw1(AT)email.com), May 23 2003

Keywords

Examples

			a(10) = 8 because the 10th Fibonacci number is 55, the 11th is 89 and the eight primes between them are 59, 61, 67, 71, 73, 79, 83 and 89.

Links

Crossrefs

Cf. A000045, A054782, A076777.
Cf. A001605, A005478.

Programs

  • Magma
    [#PrimesInInterval(Fibonacci(n-1), Fibonacci(n)): n in [2..45]]; // Vincenzo Librandi, Jul 13 2017
  • Mathematica
    lst={};Do[p=0;Do[If[PrimeQ[a],p++ ],{a,Fibonacci[n],Fibonacci[n+1]}];AppendTo[lst,p],{n,50}];lst (* Vladimir Joseph Stephan Orlovsky, Nov 23 2009 *)
  • PARI
    { a(n)= c=0; forprime(N=fibonacci(n),fibonacci(n+1),c=c+1); return(c); }

Extensions

Corrected and extended by Rick L. Shepherd, May 26 2003
a(43)-a(44) from Vincenzo Librandi, Jul 13 2017

A350701 a(n) is the number of squares strictly between Fibonacci(n) and Fibonacci(n+1).

Original entry on oeis.org

0, 0, 0, 0, 1, 0, 1, 1, 1, 2, 2, 2, 3, 4, 5, 7, 8, 11, 14, 18, 22, 29, 36, 46, 58, 75, 95, 120, 154, 195, 248, 315, 402, 511, 649, 826, 1052, 1337, 1700, 2164, 2751, 3501, 4452, 5664, 7204, 9164, 11656, 14828, 18861, 23991, 30518, 38818, 49379, 62810, 79896
Offset: 0

Views

Author

Karl-Heinz Hofmann, Jan 24 2022

Keywords

Comments

Terms a(0..3) are of course 0, because A000045(4) = 3 and A000045(5) = 5 are the first terms which are letting room for at least 1 integer.

Examples

			Strictly between Fibonacci(9) = 34 and Fibonacci(10) = 55 are the 2 squares 36 and 49. So a(9) = 2.

Links

Crossrefs

Cf. A000045, A000290, A061287, A076777.

Programs

  • PARI
    a(n)={if(n<=1, 0, sqrtint(fibonacci(n+1)-1) - sqrtint(fibonacci(n)))} \\ Andrew Howroyd, Jan 25 2022
  • Python
    from math import isqrt
from sympy import fibonacci as fi
print([0,0] + [(isqrt(fi(k+1)-1) - isqrt(fi(k))) for k in range(2, 55)])
  • Python
    from math import isqrt
from gmpy2 import fib2
def A350701(n): return 0 if n <= 1 else (lambda x:isqrt(x[0]-1)-isqrt(x[1]))(fib2(n+1)) # Chai Wah Wu, Jan 25 2022

A130973 Number of primes between successive pairs of twin primes, for a(n) > 0.

Original entry on oeis.org

1, 1, 2, 1, 4, 3, 4, 2, 1, 3, 1, 2, 3, 10, 4, 7, 4, 3, 2, 1, 2, 18, 2, 2, 17, 1, 2, 6, 9, 3, 1, 1, 1, 8, 3, 2, 15, 1, 4, 1, 1, 7, 7, 4, 4, 3, 4, 1, 1, 7, 2, 5, 1, 5, 18, 2, 5, 4, 3, 1, 5, 1, 18, 12, 2, 8, 1, 4, 2, 5, 4, 1, 1, 1, 9, 10
Offset: 1

Views

Author

Omar E. Pol, Aug 23 2007

Keywords

Comments

a(k) corresponds to the k-th term in the isolated prime sequence A007510 or A134797. a(1) corresponds to 23. a(2) corresponds to 37. a(3) corresponds to 47 and 53. - Enrique Navarrete, Jan 28 2017
Lengths of the runs of consecutive integers in A176656. - R. J. Mathar, Feb 19 2017

Links

Crossrefs

Cf. A001223, A007510 (isolated primes), A027883, A048614, A048198, A052011, A052012, A061273, A076777, A073784, A082602, A088700, A179067 (clusters of twin primes).
Showing 1-3 of 3 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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