A242950 - OEIS

%I #7 May 28 2014 02:36:20

%S 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,

%T 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,

%U 5,4,8,2,3,4,5,4,5,6,7,7

%N 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.

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

%C (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.

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

%H Zhi-Wei Sun, <a href="/A242950/b242950.txt">Table of n, a(n) for n = 1..10000</a>

%e a(11) = 1 since 11 = 2 + 9, prime(2) = 3 == 1^2 (mod 2), and prime(9) = 23 == 5 (mod 9) with 5 prime.

%e a(16) = 1 since 16 = 12 + 4, prime(12) = 37 == 1^2 (mod 12), and prime(4) = 7 == 3 (mod 4) with 3 prime.

%t SQ[n_]:=IntegerQ[Sqrt[n]]

%t s[k_]:=SQ[Mod[Prime[k],k]]

%t p[k_]:=PrimeQ[Mod[Prime[k],k]]

%t a[n_]:=Sum[Boole[s[k]&&p[n-k]],{k,2,n-2}]

%t Table[a[n],{n,1,80}]

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

%K nonn

%O 1,9

%A _Zhi-Wei Sun_, May 27 2014