OFFSET
1,2
COMMENTS
Conjecture: (i) a(n) > 0 for all n > 0.
(ii) Each positive integer can be written as 2^x + pi(y^2) + pi(z^2) with x >= 0, y > 0 and z > 0.
REFERENCES
Zhi-Wei Sun, Problems on combinatorial properties of primes, in: M. Kaneko, S. Kanemitsu and J. Liu (eds.), Number Theory: Plowing and Starring through High Wave Forms, Proc. 7th China-Japan Seminar (Fukuoka, Oct. 28 - Nov. 1, 2013), Ser. Number Theory Appl., Vol. 11, World Sci., Singapore, 2015, pp. 169-187.
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 [math.NT], 2014.
EXAMPLE
a(1) = 1 since 1 = 2^0 + 0^2 + pi(1^2).
a(2) = 2 since 2 = 2^0 + 1^2 + pi(1^2) = 2 + 0^2 + pi(1^2).
a(3) = 2 since 3 = 2^0 + 0^2 + pi(2^2) = 2 + 1^2 + pi(1^2).
MATHEMATICA
SQ[n_]:=IntegerQ[Sqrt[n]]
f[n_]:=PrimePi[n^2]
Do[r=0; Do[If[f[x]>=n, Goto[aa]]; Do[If[2^y>n-f[x], Goto[bb]]; If[SQ[n-f[x]-2^y], r=r+1], {y, 0, Log[2, n-f[x]]}]; Label[bb]; Continue, {x, 1, n}]; Label[aa]; Print[n, " ", r]; Continue, {n, 1, 100}]
CROSSREFS
KEYWORD
nonn
AUTHOR
Zhi-Wei Sun, Oct 05 2015
STATUS
approved