A236765 - OEIS

%I #16 Feb 02 2014 06:00:25

%S 0,0,0,0,0,0,1,1,1,1,1,1,2,1,1,2,2,0,2,3,1,3,2,1,2,2,3,2,4,3,2,3,2,2,

%T 3,3,2,2,2,3,4,4,3,4,4,1,3,4,2,2,5,3,3,4,4,3,1,5,3,4,3,4,5,4,3,1,5,2,

%U 6,4,3,4,2,1,5,4,7,4,4,3,1,3,1,4,4,4,2,5,6,3,6,5,5,1,4,5,5,4,3,6

%N Number of ways to write n = k^2 + m with k > 1 and m > 1 such that sigma(k^2) + prime(m) - 1 is prime, where sigma(j) denotes the sum of all positive divisors of j.

%C Conjecture: (i) If n > 6 is not equal to 18, then a(n) > 0.

%C (ii) Any integer n > 14 can be written as p + q with q > 0 such that p, p + 6 and prime(p) + sigma(q) are all prime.

%H Zhi-Wei Sun, <a href="/A236765/b236765.txt">Table of n, a(n) for n = 1..10000</a>

%p a(10) = 1 since 10 = 2^2 + 6 with sigma(2^2) + prime(6) - 1 = 7 + 13 - 1 = 19 prime.

%p a(253) = 1 since 253 = 15^2 + 28 with sigma(15^2) + prime(28) - 1 = 403 + 107 - 1 = 509 prime.

%t p[n_,k_]:=PrimeQ[DivisorSigma[1,k^2]+Prime[n-k^2]-1]

%t a[n_]:=If[n<6,0,Sum[If[p[n,k],1,0],{k,2,Sqrt[n-2]}]]

%t Table[a[n],{n,1,100}]

%Y Cf. A000040, A000203, A233544, A233654, A233793, A233864, A236548.

%K nonn

%O 1,13

%A _Zhi-Wei Sun_, Jan 30 2014