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A231555 Number of ways to write n = x + y (x, y > 0) with x*(x + 1) + F(y) prime, where F(y) denotes the y-th Fibonacci number (A000045). 8
0, 1, 2, 2, 2, 2, 3, 3, 3, 1, 2, 4, 2, 3, 4, 6, 3, 5, 1, 3, 5, 6, 6, 4, 5, 5, 4, 7, 5, 1, 5, 6, 6, 6, 6, 6, 8, 6, 5, 5, 5, 5, 6, 3, 4, 8, 9, 8, 4, 5, 8, 8, 6, 5, 9, 5, 9, 8, 8, 6, 9, 7, 8, 7, 6, 4, 8, 7, 8, 11, 6, 7, 9, 4, 5, 8, 8, 7, 10, 10, 11, 9, 3, 5, 6, 6, 4, 12, 5, 9, 12, 11, 7, 6, 7, 9, 6, 10, 5, 6 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,3

COMMENTS

Conjecture: (i) a(n) > 0 for all n > 1. Also, any integer n > 1 can be written as x + y (x, y > 0) with x + F(y) prime.

(ii) Each positive integer n not among 1, 7, 55 can be written as x + y (x, y > 0) with x*(x+1)/2 + F(y) prime. Also, any positive integer n not among 1, 10, 13, 20, 255 can be written as x + y (x, y > 0) with x^2 + F(y) prime.

We also have similar conjectures involving some second-order recurrences other than the Fibonacci sequence.

LINKS

Zhi-Wei Sun, Table of n, a(n) for n = 1..5000

EXAMPLE

a(19) = 1 since 19 = 17 + 2 with 17*18 + F(2) = 307 prime.

a(30) = 1 since 30 = 8 + 22 with 8*9 + F(22) = 17783 prime.

MATHEMATICA

a[n_]:=Sum[If[PrimeQ[x(x+1)+Fibonacci[n-x]], 1, 0], {x, 1, n-1}]

Table[a[n], {n, 1, 100}]

CROSSREFS

Cf. A000040, A000045, A231201, A228425, A228428, A228429, A228430, A228431.

Sequence in context: A224713 A168557 A194320 * A120425 A104186 A184320

Adjacent sequences:  A231552 A231553 A231554 * A231556 A231557 A231558

KEYWORD

nonn

AUTHOR

Zhi-Wei Sun, Nov 10 2013

STATUS

approved

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Last modified October 23 22:58 EDT 2019. Contains 328379 sequences. (Running on oeis4.)