OFFSET
1,1
COMMENTS
Conjecture: Let n be any positive integer. Then a(n) > 0. Moreover, one of the four consecutive numbers 5*n, 5*n+1, 5*n+2, 5*n+3 can be written as p^2+q with p and q both prime.
It seems that there are infinitely many positive integers n such that none of n, n+1, n+2, n+3, n+4 has the form p^2 + q with p and q both prime.
See also A258141 for a similar conjecture.
Note that neither 3763 nor 5443 can be written as floor((p^2+q)/4) with p and q both prime.
LINKS
Zhi-Wei Sun, Table of n, a(n) for n = 1..10000
Zhi-Wei Sun, Natural numbers represented by floor(x^2/a) + floor(y^2/b) + floor(z^2/c), arXiv:1504.01608 [math.NT], 2015.
EXAMPLE
a(1) = 3 since 1 = floor((2^2+2)/5) = floor((2^2+3)/5) = floor(2^2+5)/5) with 2, 3, 5 all prime.
a(2) = 4 since 2 = floor((2^2+7)/5) = floor((3^2+2)/5) = floor((3^2+3)/5) = floor((3^2+5)/5) with 2, 3, 5, 7 all prime.
MATHEMATICA
Do[m=0; Do[If[PrimeQ[5n+r-Prime[k]^2], m=m+1], {r, 0, 4}, {k, 1, PrimePi[Sqrt[5n+r]]}]; Print[n, " ", m]; Continue, {n, 1, 80}]
CROSSREFS
KEYWORD
nonn
AUTHOR
Zhi-Wei Sun, May 22 2015
STATUS
approved