A233359 a(n) = |{0 < k < n: L(k) + q(n-k) is prime}|, where L(k) is the k-th Lucas number (A000204), and q(.) is the strict partition function (A000009).

0, 1, 1, 2, 3, 1, 2, 4, 2, 2, 3, 3, 2, 4, 3, 5, 1, 4, 5, 3, 1, 3, 3, 7, 3, 3, 4, 5, 2, 2, 9, 2, 4, 4, 9, 2, 6, 6, 6, 3, 3, 1, 5, 7, 4, 4, 5, 7, 4, 9, 5, 6, 4, 1, 5, 6, 11, 9, 4, 2, 5, 5, 4, 6, 8, 9, 12, 3, 7, 5, 4, 10, 6, 7, 6, 3, 5, 8, 4, 4, 4, 4, 7, 7, 5, 1, 4, 9, 7, 4, 8, 7, 6, 5, 2, 3, 7, 11, 5, 5

Offset

1,4

Comments

Conjecture: a(n) > 0 for all n > 1.

We have verified this for n up to 60000.

Note that for n = 19976 there is no k = 0,...,n such that F(k) + q(n-k) is prime, where F(0), F(1), ... are the Fibonacci numbers.

Links

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

Example

a(7) = 2 since L(1) + q(6) = 1 + 4 = 5 and L(6) + q(1) = 18 + 1 = 19 are both prime.
a(17) = 1 since L(13) + q(4) = 521 + 2 = 523 is prime.
a(21) = 1 since L(5) + q(16) = 11 + 32 = 43 is prime.
a(42) = 1 since L(22) + q(20) = 39603 + 64 = 39667 is prime.
a(54) = 1 since L(8) + q(46) = 47 + 2304 = 2351 is prime.
a(86) = 1 since L(67) + q(19) = 100501350283429 + 54 = 100501350283483 is prime.

Mathematica

a[n_]:=Sum[If[PrimeQ[LucasL[k]+PartitionsQ[n-k]], 1, 0], {k, 1, n-1}]
Table[a[n], {n, 1, 100}]

Crossrefs

Cf. A000009, A000032, A000040, A000204, A233307, A233346, A233360.

Sequence in context: A329631 A239304 A072193 * A279345 A097966 A177993

Adjacent sequences: A233356 A233357 A233358 * A233360 A233361 A233362

Keyword

nonn

Author

Zhi-Wei Sun, Dec 08 2013

Status

approved