A233567 - OEIS

%I #6 Dec 13 2013 18:18:28

%S 0,0,1,1,0,1,1,2,2,2,3,1,3,2,4,2,3,4,3,4,5,3,5,2,6,4,3,4,5,2,1,2,3,5,

%T 5,1,3,3,4,3,3,7,6,4,7,2,5,5,5,5,3,7,4,7,4,6,5,3,5,6,6,5,5,8,9,6,7,5,

%U 6,5,7,7,5,8,7,6,6,6,8,8,5,8,11,3,7,6,7,8,7,1,8,5,6,9,10,8,9,12,8,6

%N Number of ways to write n = p + q (q > 0) with p and p^4 + phi(q)^4 both prime, where phi(.) is Euler's totient function (A000010).

%C Conjecture: If n > 2 is not equal to 5, then we have a(n) > 0, also there is a prime p < n with p^2 + phi(n-p)^2 prime.

%C We have verified this for n up to 10^7. The first assertion in the conjecture implies that there are infinitely many primes of the form p^4 + q^4, where p is a prime and q is a positive integer.

%H Zhi-Wei Sun, <a href="/A233567/b233567.txt">Table of n, a(n) for n = 1..10000</a>

%e a(7) = 1 since 7 = 3 + 4 with 3 and 3^4 + phi(4)^4 = 81 + 16 = 97 both prime.

%e a(12) = 1 since 12 = 7 + 5 with 7 and 7^4 + phi(5)^4 = 7^4 + 4^4 = 2657 both prime.

%e a(31) = 1 since 31 = 23 + 8 with 23 and 23^4 + phi(8)^4 = 23^4 + 4^4 = 280097 both prime.

%e a(36) = 1 since 36 = 3 + 33 with 3 and 3^4 + phi(33)^4 = 3^4 + 20^4 = 160081 both prime.

%e a(90) = 1 since 90 = 79 + 11 with 79 and 79^4 + phi(11)^4 = 79^4 + 10^4 = 38960081 both prime.

%t a[n_]:=Sum[If[PrimeQ[Prime[k]^4+EulerPhi[n-Prime[k]]^4],1,0],{k,1,PrimePi[n-1]}]

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

%Y Cf. A000010, A000040, A002645, A233542, A233544, A233547, A233549, A233566.

%K nonn

%O 1,8

%A _Zhi-Wei Sun_, Dec 13 2013