A239214 a(n) = |{0 < k < n: p(k)*p(n)*(p(n)+1) - 1 is prime}|, where p(.) is the partition function (A000041).

0, 1, 2, 3, 1, 3, 3, 2, 3, 3, 5, 4, 4, 3, 3, 6, 2, 4, 5, 4, 1, 2, 3, 6, 6, 6, 2, 4, 6, 9, 2, 7, 8, 6, 6, 2, 2, 2, 10, 4, 4, 7, 5, 7, 1, 4, 9, 9, 9, 4, 6, 8, 7, 8, 6, 4, 13, 10, 3, 6, 10, 7, 13, 12, 12, 8, 6, 8, 5, 11, 5, 3, 4, 5, 11, 7, 6, 12, 16, 4

Offset

1,3

Comments

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

(ii) For each n = 2, 3, ... there is a positive integer k < n with p(k)*p(n)*(p(n)-1) + 1 prime. If n > 2, then p(k)*p(n)*(p(n)-1)-1 is prime for some 0 < k < n.

(iii) For any n > 1, there is a positive integer k < n with 2*p(k)*p(n)*A000009(n)*A047967(n) + 1 prime.

We have verified that a(n) > 0 for all n = 2, ..., 10^5.

Links

Zhi-Wei Sun, Table of n, a(n) for n = 1..10000

Zhi-Wei Sun, Problems on combinatorial properties of primes, arXiv:1402.6641, 2014.

Example

a(2) = 1 since p(1)*p(2)*(p(2)+1) - 1 = 1*2*3 - 1 = 5 is prime.
a(5) = 1 since p(3)*p(5)*(p(5)+1) - 1 = 3*7*8 - 1 = 167 is prime.
a(21) = 1 since p(10)*p(21)*(p(21)+1) - 1 = 42*792*793 - 1 = 26378351 is prime.
a(45) = 1 since p(20)*p(45)*(p(45)+1) - 1 = 627*89134*89135 - 1 = 4981489349429 is prime.

Mathematica

p[n_]:=PartitionsP[n]
f[n_]:=p[n]*(p[n]+1)
a[n_]:=Sum[If[PrimeQ[p[k]*f[n]-1], 1, 0], {k, 1, n-1}]
Table[a[n], {n, 1, 80}]

Crossrefs

Cf. A000040, A000041, A238457, A238509, A238516, A238393, A239207, A239209.

Sequence in context: A230294 A104483 A080717 * A200181 A121062 A045831

Adjacent sequences: A239211 A239212 A239213 * A239215 A239216 A239217

Keyword

nonn

Author

Zhi-Wei Sun, Mar 12 2014

Status

approved