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.

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, 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, 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

Offset: 1

Author

Zhi-Wei Sun, Nov 22 2013

Keywords

• nonn

Comments

Conjecture: (i) a(n) > 0 for all n > 1, and a(n) = 1 only for n = 2, 3, 4, 29, 70, 105.

(ii) Any integer n > 164 can be written as x + y (x, y > 0) with sigma(x) + sigma(y) prime.

Examples

			a(2) = 1 since 2 = 1 + 1 with sigma(1) + phi(1) = 2 prime.
a(29) = 1 since 29 = 1 + 28 with sigma(1) + phi(28) = 13 prime.
a(70) = 1 since 70 = 9 + 61 with sigma(9) + phi(61) = 73 prime.
a(105) = 1 since 105 = 4 + 101 with sigma(4) + phi(101) = 107 prime.
		

Links

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

Crossrefs

Cf. A000010, A000040, A000203.

Programs

Mathematica

f[n_]:=f[n]=Sum[If[Mod[n,d]==0,d,0],{d,1,n}]
a[n_]:=a[n]=Sum[If[PrimeQ[f[k]+EulerPhi[n-k]],1,0],{k,1,n/2}]
Table[a[n],{n,1,100}]