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 1-4 of 4 results.

A132145 Numbers that can be presented as a sum of a prime number and a Fibonacci number (0 is considered to be a Fibonacci number).

Original entry on oeis.org

2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72
Offset: 1

Views

Author

Tanya Khovanova, Aug 12 2007

Keywords

Comments

This sequence is the union of prime numbers and sequence A132147. It is also the complement of A132144.
Lee shows that the set of the numbers that are the sum of a prime and a Fibonacci number has positive lower asymptotic density. [Jonathan Vos Post, Nov 02 2010]

Examples

			11 = 3+8, the sum of a prime number (3) and a Fibonacci number (8).

Links

Crossrefs

Cf. A132144, A132147.

Programs

  • Maple
    N:= 1000: # for all entries <= N
Primes:= select(isprime,{$1..N}):
phi:= (1+sqrt(5))/2:
Fibs:= {seq(combinat:-fibonacci(i),i=0..floor(log[phi]((N+1)*sqrt(5))))}:
sort(convert(select(`<=`,{seq(seq(f+p,f=Fibs),p=Primes)},N),list)); # Robert Israel, Aug 03 2015
  • Mathematica
    Take[Union[Flatten[Table[Fibonacci[n] + Prime[k], {n, 70}, {k, 70}]], Table[Prime[k], {k, 70}]], 70]

A132146 Numbers that can't be presented as a sum of a prime number and a Fibonacci number (0 is not considered to be a Fibonacci number).

Original entry on oeis.org

1, 2, 17, 29, 35, 59, 83, 89, 119, 125, 127, 177, 179, 208, 209, 221, 239, 255, 269, 287, 299, 329, 331, 353, 359, 363, 389, 416, 419, 449, 479, 485, 509, 515, 519, 535, 539, 547, 551, 561, 567, 569, 599, 637, 659, 673, 697, 705, 718, 733, 739, 755, 768, 779
Offset: 1

Views

Author

Tanya Khovanova, Aug 12 2007

Keywords

Comments

This sequence contains A132144 as a subsequence and is the complement of A132147.

Examples

			The smallest prime number is 2, the smallest Fibonacci number is 1; hence 1 and 2 can't be presented as a sum of a prime number and a Fibonacci number.

Links

Programs

  • Mathematica
    Complement[Range[1000], Take[Union[Flatten[Table[Fibonacci[n] + Prime[k], {n, 700}, {k, 700}]]], 1000]]

A131511 All possible products of prime and Fibonacci numbers.

Original entry on oeis.org

0, 2, 3, 4, 5, 6, 7, 9, 10, 11, 13, 14, 15, 16, 17, 19, 21, 22, 23, 24, 25, 26, 29, 31, 33, 34, 35, 37, 38, 39, 40, 41, 42, 43, 46, 47, 51, 53, 55, 56, 57, 58, 59, 61, 62, 63, 65, 67, 68, 69, 71, 73, 74, 79, 82, 83, 85, 86, 87, 88, 89, 91, 93, 94, 95, 97, 101, 102, 103, 104, 105
Offset: 1

Views

Author

Tanya Khovanova, Aug 14 2007

Keywords

Comments

This sequence contains all prime numbers as a subsequence because 1 is a Fibonacci number. Similarly it contains all even semiprimes.

Examples

			8 is not in this sequence because the only way to represent 8 as a product of a prime and some number is 2*4 and 4 is not a Fibonacci number.
105 is in this sequence because 105 = 3*21 and 3 is a prime number and 21 is a Fibonacci number.

Links

Crossrefs

Cf. A000045, A001358, A049997, A132147.

Programs

  • Mathematica
    Take[Union[Flatten[Table[Fibonacci[n]*Prime[k], {n, 70}, {k, 70}]]], 70]
  • PARI
    isfib(n) = my(k=n^2); k+=(k+1)<<2; issquare(k) || (n>0 && issquare(k-8));
isok(n) = {if (n==0, return (1)); my(f=factor(n)); for (k=1, #f~, p = f[k, 1]; if (isfib(n/p), return (1)););} \\ Michel Marcus, Apr 19 2018

A367187 Numbers which are the sum of a prime number and a Fibonacci number of index >1 in at least two ways.

Original entry on oeis.org

4, 5, 6, 7, 8, 10, 12, 13, 14, 15, 16, 18, 19, 20, 21, 22, 24, 25, 26, 28, 30, 31, 32, 34, 36, 37, 38, 39, 40, 42, 44, 45, 46, 48, 49, 50, 51, 52, 54, 55, 56, 57, 58, 60, 61, 62, 63, 64, 66, 68, 69, 72, 74, 75, 76, 78, 80, 81, 82, 84, 86, 87, 88, 91, 92, 94, 96, 98, 100
Offset: 1

Views

Author

Yuda Chen, Nov 08 2023

Keywords

Examples

			4 is a term since 4 = 1+3 = 2+2.
5 is a term since 5 = 2+3 = 3+2.
57 is a term since 57 = 2+55 = 23+34.

Crossrefs

Cf. A000040, A000045, A367186.
Subsequence of A132147.

Programs

  • PARI
    isfib(n) = if (n>0, my(k=n^2); k+=(k+1)<<2; issquare(k) || issquare(k-8));
isok(k) = sum(i=1, primepi(k), isfib(k-prime(i))) > 1; \\ Michel Marcus, Nov 09 2023
Showing 1-4 of 4 results.

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

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