A242950 Number of ordered ways to write n = k + m with k > 1 and m > 1 such that the least nonnegative residue of prime(k) modulo k is a square and the least nonnegative residue of prime(m) modulo m is a prime.

0, 0, 0, 0, 1, 1, 0, 1, 3, 2, 1, 1, 3, 4, 4, 1, 3, 5, 4, 4, 4, 3, 3, 3, 3, 4, 4, 4, 3, 5, 2, 5, 3, 5, 3, 6, 3, 7, 4, 6, 5, 7, 5, 9, 7, 6, 4, 6, 5, 9, 5, 6, 8, 7, 8, 5, 8, 5, 8, 4, 8, 6, 7, 4, 7, 4, 6, 4, 5, 4, 8, 2, 3, 4, 5, 4, 5, 6, 7, 7

Offset: 1

Author

Zhi-Wei Sun, May 27 2014

Keywords

• nonn

Comments

Conjecture: (i) a(n) > 0 for all n > 7.

(ii) Any integer n > 9 can be written as k + m with k > 1 and m > 1 such that the least nonnegative residue of prime(k) modulo k and the least nonnegative residue of prime(m) modulo m are both prime.

We have verified a(n) > 0 for all n = 8, ..., 10^8.

Examples

			a(11) = 1 since 11 = 2 + 9, prime(2) = 3 == 1^2 (mod 2), and prime(9) = 23 == 5 (mod 9) with 5 prime.
a(16) = 1 since 16 = 12 + 4, prime(12) = 37 == 1^2 (mod 12), and prime(4) = 7 == 3 (mod 4) with 3 prime.
		

Links

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

Crossrefs

Cf. A000290, A000040, A242425, A242748, A242753.

Programs

Mathematica

SQ[n_]:=IntegerQ[Sqrt[n]]
s[k_]:=SQ[Mod[Prime[k],k]]
p[k_]:=PrimeQ[Mod[Prime[k],k]]
a[n_]:=Sum[Boole[s[k]&&p[n-k]],{k,2,n-2}]
Table[a[n],{n,1,80}]