OFFSET
1,5
COMMENTS
Conjecture: (i) a(n) > 0 for all n > 3.
(ii) If n > 15, then p(n+k) - 1 is prime for some k = 1, ..., n.
(iii) If n > 38, then p(n+k) is prime for some k = 1, ..., n.
The conjecture implies that there are infinitely many positive integers m with p(m) + 1 (or p(m) - 1, or p(m)) prime.
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(4) = 1 since p(4+4) + 1 = 22 + 1 = 23 is prime.
a(8) = 2 since p(8+1) + 1 = 31 and p(8+2) + 1 = 43 are both prime.
a(11) = 1 since p(11+8) + 1 = 491 is prime.
MATHEMATICA
p[n_]:=PartitionsP[n]
a[n_]:=Sum[If[PrimeQ[p[n+k]+1], 1, 0], {k, 1, n}]
Table[a[n], {n, 1, 80}]
CROSSREFS
KEYWORD
nonn
AUTHOR
Zhi-Wei Sun, Mar 13 2014
STATUS
approved