OFFSET
1,5
COMMENTS
Conjecture: (i) a(n) > 0 for all n > 3.
(ii) For any integer n > 2, |n - p(k)| is prime for some k = 1,...,n. Also, for any positive integer n not equal to 7, n + p(k) is prime for some k = 1,...,n.
We have verified part (i) of the conjecture for all n = 4, 5, ..., 2*10^7.
LINKS
Zhi-Wei Sun, Table of n, a(n) for n = 1..3000
Z.-W. Sun, On a^n+ bn modulo m, arXiv preprint arXiv:1312.1166 [math.NT], 2013-2014.
EXAMPLE
a(6) = 3 since 6 = 3 + p(1) + p(2) = 2 + p(1) + p(3) = 2 + p(2) + p(2) with 2 and 3 prime.
MATHEMATICA
PQ[n_]:=n>1&&PrimeQ[n]
a[n_]:=Sum[If[PQ[n-PartitionsP[m]-PartitionsP[k]], 1, 0], {m, 1, n}, {k, 1, m}]
Table[a[n], {n, 1, 100}]
CROSSREFS
KEYWORD
nonn
AUTHOR
Zhi-Wei Sun, Nov 24 2013
STATUS
approved