A263100 Number of ordered pairs (k, m) with k > 0 and m > 0 such that n = pi(k^2) + pi(m^2/2), where pi(x) denotes the number of primes not exceeding x.

1, 2, 1, 3, 2, 3, 3, 2, 4, 2, 6, 2, 5, 2, 5, 4, 4, 4, 4, 5, 3, 5, 5, 4, 4, 6, 6, 1, 7, 4, 6, 4, 4, 7, 6, 4, 5, 5, 5, 6, 6, 4, 6, 3, 7, 6, 5, 6, 6, 6, 5, 5, 6, 4, 7, 8, 4, 3, 10, 2, 6, 6, 6, 6, 7, 5, 5, 9, 3, 6, 8, 6, 7, 5, 5, 6, 7, 7, 8, 3, 9, 3, 10, 2, 7, 9, 7, 2, 7, 8, 5, 8, 4, 6, 9, 5, 7, 6, 5, 7

Offset: 1

Author

Zhi-Wei Sun, Oct 09 2015

Keywords

• nonn

Comments

Conjecture: (i) a(n) > 0 for all n > 0, and a(n) = 1 only for n = 1, 3, 28.

(ii) Any integer n > 0 can be written as pi(k^2) + pi((m^2+1)/2) with k and m positive integers.

(iii) Each n = 1,2,3,... can be written as pi(k^2/2) + pi((m^2+1)/2) with k and m positive integers.

See also A262995, A262999, A263001 and A263020 for similar conjectures.

Examples

			a(1) = 1 since 1 = 0 + 1 = pi(1^2) + pi(2^2/2).
a(3) = 1 since 3 = 2 + 1 = pi(2^2) + pi(2^2/2).
a(28) = 1 since 28 = 11 + 17 = pi(6^2) + pi(11^2/2).
		

Links

• Zhi-Wei Sun, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A000290, A000720, A038107, A262995, A262999, A263001, A263020.

Programs

Mathematica

s[n_]:=s[n]=PrimePi[n^2]
t[n_]:=t[n]=PrimePi[n^2/2]
Do[r=0; Do[If[s[k]>n, Goto[bb]]; Do[If[t[j]>n-s[k], Goto[aa]]; If[t[j]==n-s[k], r=r+1]; Continue, {j, 1, n-s[k]+1}]; Label[aa]; Continue, {k, 1, n}];
Label[bb]; Print[n, " ", r]; Continue, {n,1,100}]