A132144 -id:A132144 - cp's OEIS frontend

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).

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

Tanya Khovanova, Aug 12 2007

nonn

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]

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

T. D. Noe, Table of n, a(n) for n=1..1000
K. S. Enoch Lee, On the sum of a prime and a Fibonacci number, arXiv:1011.0173 [math.NT], 2010.

Cf. A132144, A132147.

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

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

Tanya Khovanova, Aug 12 2007

nonn

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

			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.

T. D. Noe, Table of n, a(n) for n=1..10000

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

A168383 Numbers expressible as the sum of a prime and a Fibonacci number in only one way, and such that the prime and Fibonacci number have the same number of decimal digits.

Original entry on oeis.org

2, 9, 65, 77, 93, 95, 123, 323, 335, 343, 377, 395, 415, 425, 437, 527, 545, 553, 583, 586, 670, 700, 715, 723, 726, 731, 749, 783, 801, 804, 833, 838, 849, 851, 901, 903, 905, 906, 923, 957, 959, 964, 965, 1003, 1078, 1081, 1113, 1115

Offset: 1

Jason Earls, Nov 24 2009

1 = Fibonacci(1) = Fibonacci(2), so cases where the Fibonacci number is 1 are counted as two ways. Also, if Fibonacci(i) and Fibonacci(j) are both primes (with i <> j), Fibonacci(i) + Fibonacci(j) and Fibonacci(j) + Fibonacci(i) are counted as two ways. - Robert Israel, Aug 22 2024

			In the decomposition of 1081, the prime and Fibonacci both have three digits: 1081 = 144 + 937.

J. Earls, "Fibonacci Prime Decompositions," Mathematical Bliss, Pleroma Publications, 2009, pages 76-79. ASIN: B002ACVZ6O

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

Cf. A000045, A132144, A375642. Contained in A375643.

filter:= proc(n) local f,i,d,state;
  state:= 0;
  for i from 0 do
    f:= combinat:-fibonacci(i);
    if f >= n then return (state = 1) fi;
    if isprime(n-f) then
      state:= state+1;
      if state = 2 then return false fi;
      if f = 0 then d:= 1 else d:= 1+ilog10(f) fi;
      if 1+ilog10(n-f) <> d then return false fi;
    fi
  od;
end proc:
select(filter, [$1..2000]); # Robert Israel, Aug 22 2024

Definition clarified by Robert Israel, Aug 22 2024

A375642 a(n) is the number of i for which n - Fibonacci(i) is prime.

Original entry on oeis.org

0, 1, 3, 3, 3, 3, 3, 4, 1, 3, 2, 3, 3, 3, 3, 3, 1, 4, 3, 4, 2, 2, 2, 5, 2, 3, 1, 2, 1, 3, 3, 5, 1, 3, 0, 3, 3, 3, 3, 2, 2, 4, 2, 5, 3, 2, 2, 3, 2, 3, 2, 2, 2, 3, 2, 2, 2, 3, 1, 4, 3, 5, 2, 3, 1, 3, 2, 4, 2, 1, 2, 5, 2, 6, 3, 2, 1, 2, 2, 4, 3, 2, 1, 5, 1, 3, 2, 2, 1, 2, 3, 5, 1, 3, 1, 3, 2, 3, 1

Offset: 1

Robert Israel, Aug 22 2024

nonn

			a(5) = 3 because 5 - Fibonacci(0) = 5, 5 - Fibonacci(3) = 3 and 5 - Fibonacci(4) = 2 are prime.

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

Cf. A000045, A108852, A132144, A168383, A169791.

fcount:= proc(n) local f,i,d,c;
  c:= 0;
  for i from 0 do
    f:= combinat:-fibonacci(i);
    if f >= n then return c fi;
    if isprime(n-f) then
      c:= c+1;
    fi
  od;
end proc:
map(f, [$1..200]);

a[n_]:=Sum[Boole[PrimeQ[n-Fibonacci[i]]],{i,Select[Range[0,n],n>Fibonacci[#]&]}]; Array[a,99] (* Stefano Spezia, Aug 23 2024 *)

A168382 Least number k having n distinct representations as the sum of a nonzero Fibonacci number and a prime.

Original entry on oeis.org

3, 4, 8, 24, 74, 444, 1600, 15684, 29400, 50124, 259224, 5332128, 11110428, 50395440, 451174728, 1296895890, 13314115434, 32868437466, 326585290794, 4788143252148

Offset: 1

Jason Earls, Nov 24 2009

The meaning of "distinct" is the following: we count ordered index pairs (i,j) with k = Fibonacci(i) + prime(j), i > 1, j >= 1.
Fibonacci(1) + prime(4) = Fibonacci(2) + prime(4) = Fibonacci(4) + prime(3) = Fibonacci(5) + prime(2) = 8 are three "distinct" representations of k=8, because Fibonacci(1) = Fibonacci(2) is treated as indistinguishable, whereas Fibonacci(4) = prime(2) are distinguishable based on the ordering in the indices (ordering in the sum): k = 1+7 = 3+5 = 5+3.
a(17) > 10^10. [Donovan Johnson, May 17 2010]

			15684 is the least number having eight distinct representations due to the following sums: 1 + 15683 = 5 + 15679 = 13 + 15671 = 55 + 15629 = 233 + 15451 = 377 + 15307 = 1597 + 14087 = 4181 + 11503.

J. Earls, "Fibonacci Prime Decompositions," Mathematical Bliss, Pleroma Publications, 2009, pages 76-79. ASIN: B002ACVZ6O

Cf. A132144, A169790, A169791.

Two more terms from R. J. Mathar, Feb 07 2010
a(7) corrected by Jon E. Schoenfield, May 14 2010
Edited by R. J. Mathar, May 14 2010
a(11)-a(14) from Max Alekseyev, May 15 2010
a(15)-a(16) from Donovan Johnson, May 17 2010
a(17) from Chai Wah Wu, Sep 04 2018
a(18)-a(20) from Giovanni Resta, Dec 10 2019

A214412 Numbers that can't be expressed as the sum of a Fibonacci number and a square of a positive integer.

Original entry on oeis.org

0, 8, 13, 15, 20, 23, 31, 32, 34, 40, 42, 45, 47, 48, 53, 55, 58, 60, 61, 63, 68, 73, 74, 75, 76, 78, 79, 87, 88, 92, 95, 96, 97, 99, 106, 107, 109, 110, 111, 112, 116, 117, 118, 120, 127, 128, 130, 131, 132, 133, 135, 137, 139, 140, 141, 143, 150, 151, 154, 156

Offset: 1

Alex Ratushnyak, Jul 16 2012

0 is considered to be a Fibonacci number.

David Radcliffe, Table of n, a(n) for n = 1..1000

Cf. A000045, A132144, A214410.

q:= proc(n) local f,g; f,g:= 0,1;
      do if f>=n       then return true
       elif issqr(n-f) then return false
       else f,g:= g,f+g
      fi od
    end:
select(q, [$0..200])[];  # Alois P. Heinz, May 22 2021

nn = 156; sq = Range[Sqrt[nn]]^2; fb = {}; i = 0; While[f = Fibonacci[i];  f < nn, i++; AppendTo[fb, f]]; fb = Union[fb]; Complement[Range[0, nn], Union[Flatten[Outer[Plus, sq, fb]]]] (* T. D. Noe, Jul 31 2012 *)

prpr = 0
prev = 1
fib = [0]*100
for n in range(100):
    fib[n] = prpr
    curr = prpr+prev
    prpr = prev
    prev = curr
#print fib[n]
for n in range(777):
    i = 1
    yes = 0
    while i*i<=n:
        r = n - i*i
        if r in fib:
            yes = 1
            break
        i += 1
    if yes==0:
        print(n, end=', ')

A375643 Numbers that are the sum of a prime and a Fibonacci number in exactly one way.

Original entry on oeis.org

2, 9, 17, 27, 29, 33, 59, 65, 70, 77, 83, 85, 89, 90, 93, 95, 99, 121, 123, 124, 127, 129, 133, 143, 145, 146, 153, 155, 166, 169, 174, 179, 188, 189, 190, 195, 203, 210, 217, 219, 222, 237, 239, 243, 249, 258, 261, 267, 269, 289, 297, 302, 303, 305, 308, 309, 310, 321, 323, 327, 331, 333, 335

Offset: 1

Robert Israel, Aug 22 2024

nonn

Numbers k such that k - A000045(i) is prime for exactly one i >= 0.

1 = Fibonacci(1) = Fibonacci(2), so cases where the Fibonacci number is 1 are counted as two ways. Also, if Fibonacci(i) and Fibonacci(j) are both primes (with i <> j), Fibonacci(i) + Fibonacci(j) and Fibonacci(j) + Fibonacci(i) are counted as two ways.

			a(5) = 29 is a term because 29 - Fibonacci(i) is prime only for i = 0.

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

Cf. A000045, A132144, A375642. Contains A168383.

filter:= proc(n) local f,i,d,state;
  state:= 0;
  for i from 0 do
    if i = 2 then next fi;
    f:= combinat:-fibonacci(i);
    if f >= n then return (state = 1) fi;
    if isprime(n-f) then
      state:= state+1;
      if state = 2 then return false fi;
    fi
  od;
end proc:
select(filter, [$1..1000]);

A214410 Numbers that can't be expressed as the sum of a square and a Fibonacci number.

Original entry on oeis.org

15, 20, 23, 31, 32, 40, 42, 45, 47, 48, 53, 58, 60, 61, 63, 68, 73, 74, 75, 76, 78, 79, 87, 88, 92, 95, 96, 97, 99, 106, 107, 109, 110, 111, 112, 116, 117, 118, 120, 127, 128, 130, 131, 132, 133, 135, 137, 139, 140, 141, 143, 150, 151, 154, 156, 158, 159, 161

Offset: 1

Alex Ratushnyak, Jul 16 2012

0 is considered to be a Fibonacci number.

			17 = 16+1, 16 is a square and 1 is a Fibonacci number, so 17 is not in the sequence.

Cf. A000045, A000290, A132144, A214412.

q:= proc(n) local f,g; f,g:= 0,1;
      do if f>n        then return true
       elif issqr(n-f) then return false
       else f,g:= g,f+g
      fi od
    end:
select(q, [$0..200])[];  # Alois P. Heinz, May 22 2021

nn = 161; sq = Range[0, Sqrt[nn]]^2; fb = {}; i = 0; While[f = Fibonacci[i];  f < nn, i++; AppendTo[fb, f]]; fb = Union[fb]; Complement[Range[0, nn], Union[Flatten[Outer[Plus, sq, fb]]]] (* T. D. Noe, Jul 31 2012 *)

prpr = 0
prev = 1
fib = [0]*100
for n in range(100):
    fib[n] = prpr
    curr = prpr+prev
    prpr = prev
    prev = curr
for n in range(1234):
    i = yes = 0
    while i*i<=n:
        r = n - i*i
        if r in fib:
            yes = 1
            break
        i += 1
    if yes==0:
        print(n, end=',')

from sympy import fibonacci
from itertools import count, takewhile
def aupto(lim):
  fbs = list(takewhile(lambda x: x<=lim, (fibonacci(i) for i in count(0))))
  sqs = list(takewhile(lambda x: x<=lim, (i*i for i in count(0))))
  return sorted(set(range(1, lim+1)) - set(f+s for f in fbs for s in sqs))
print(aupto(161)) # Michael S. Branicky, May 22 2021

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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).

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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).

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A168383 Numbers expressible as the sum of a prime and a Fibonacci number in only one way, and such that the prime and Fibonacci number have the same number of decimal digits.

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A375642 a(n) is the number of i for which n - Fibonacci(i) is prime.

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A168382 Least number k having n distinct representations as the sum of a nonzero Fibonacci number and a prime.

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A214412 Numbers that can't be expressed as the sum of a Fibonacci number and a square of a positive integer.

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A375643 Numbers that are the sum of a prime and a Fibonacci number in exactly one way.

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A214410 Numbers that can't be expressed as the sum of a square and a Fibonacci number.

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