A169791 -id:A169791 - cp's OEIS frontend

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

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

A169790 Least number k having n unordered partitions into a nonzero Fibonacci number and a prime.

Original entry on oeis.org

3, 4, 10, 24, 74, 444, 1614, 15684, 29400, 50124, 259224, 5332128, 11110428, 50395440, 451174728, 1296895890

Offset: 1

R. J. Mathar and Jon E. Schoenfield, May 14 2010

Variant of A168382.
Fibonacci(1) + prime(4) = Fibonacci(2) + prime(4) = Fibonacci(4) + prime(3) = Fibonacci(5) + prime(2) = 8 are two "distinct" representations of k=8, because Fibonacci(1) = Fibonacci(2) = 1 is treated as indistinguishable, and Fibonacci(4) = prime(2) = 3 are also indistinguishable: k = 1+7 = 3+5.
This matters because of the existence of Fibonacci primes (see A005478).
a(17) > 10^10. [Donovan Johnson, May 17 2010]

			1+443 = 5+439 = 13+431 = 55+389 = 233+211 = 377+67 are n=6 distinct representations of 444.

Cf. A168382, A169791.

a(8)-a(14) from Max Alekseyev, May 15 2010
a(15)-a(16) from Donovan Johnson, May 17 2010
Prime index in the comment corrected by R. J. Mathar, Jun 02 2010

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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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A169790 Least number k having n unordered partitions into a nonzero Fibonacci number and a prime.

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