cp's OEIS Frontend

A232270 Number of ways to write n = x + y (0 < x <= y) with sigma(x) + phi(y) prime, where sigma(x) is the sum of all (positive) divisors of x and phi is Euler's totient function.

%I A232270 #7 Nov 22 2013 03:17:01
%S A232270 0,1,1,1,2,2,2,2,2,2,2,3,3,3,2,2,2,5,3,3,2,3,3,3,2,3,4,3,1,2,3,4,2,2,
%T A232270 3,3,3,6,3,3,4,5,5,4,2,2,4,5,3,5,3,3,2,4,2,6,5,2,6,3,6,5,2,6,5,4,4,6,
%U A232270 3,1,5,4,6,6,2,8,6,5,4,5,3,5,4,6,2,3,8,3,5,6,4,5,4,3,5,3,4,7,6,6
%N A232270 Number of ways to write n = x + y (0 < x <= y) with sigma(x) + phi(y) prime, where sigma(x) is the sum of all (positive) divisors of x and phi is Euler's totient function.
%C A232270 Conjecture:  (i) a(n) > 0 for all n > 1, and a(n) = 1 only for n = 2, 3, 4, 29, 70, 105.
%C A232270 (ii) Any integer n > 164 can be written as x + y (x, y > 0) with sigma(x) + sigma(y) prime.
%H A232270 Zhi-Wei Sun, <a href="/A232270/b232270.txt">Table of n, a(n) for n = 1..10000</a>
%e A232270 a(2) = 1 since 2 = 1 + 1 with sigma(1) + phi(1) = 2 prime.
%e A232270 a(29) = 1 since 29 = 1 + 28 with sigma(1) + phi(28) = 13 prime.
%e A232270 a(70) = 1 since 70 = 9 + 61 with sigma(9) + phi(61) = 73 prime.
%e A232270 a(105) = 1 since 105 = 4 + 101 with sigma(4) + phi(101) = 107 prime.
%t A232270 f[n_]:=f[n]=Sum[If[Mod[n,d]==0,d,0],{d,1,n}]
%t A232270 a[n_]:=a[n]=Sum[If[PrimeQ[f[k]+EulerPhi[n-k]],1,0],{k,1,n/2}]
%t A232270 Table[a[n],{n,1,100}]
%Y A232270 Cf. A000010, A000040, A000203.
%K A232270 nonn
%O A232270 1,5
%A A232270 _Zhi-Wei Sun_, Nov 22 2013