OFFSET
1,3
COMMENTS
Conjecture: (i) a(n) > 0 for all n > 0, and a(n) = 1 only for n = 1, 2, 13.
(ii) If n > 1, then 2*p(k)*p(n) - 1 is prime for some 0 < k < n.
(iii) For any integer n > 0, p(k)*(p(n)+1) + 1 is prime for some k = 1, ..., n.
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 2*p(1)*p(2) + 1 = 2*1*2 + 1 = 5 is prime.
a(13) = 1 since 2*p(3)*p(13) + 1 = 2*3*101 + 1 = 607 is prime.
MATHEMATICA
p[n_, k_]:=PrimeQ[2*PartitionsP[n]*PartitionsP[k]+1]
a[n_]:=Sum[If[p[n, k], 1, 0], {k, 1, n}]
Table[a[n], {n, 1, 80}]
CROSSREFS
KEYWORD
nonn
AUTHOR
Zhi-Wei Sun, Mar 01 2014
STATUS
approved