A236374 a(n) = |{0 < k < n: m = phi(k)/2 + phi(n-k)/8 is an integer with 2^(m-1)*phi(m) - 1 prime}|, where phi(.) is Euler's totient function.

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 2, 0, 2, 1, 2, 2, 2, 4, 2, 1, 0, 2, 1, 2, 3, 3, 3, 4, 2, 2, 2, 3, 5, 3, 4, 4, 1, 1, 2, 3, 7, 4, 3, 5, 3, 3, 2, 4, 5, 4, 3, 4, 3, 2, 6, 7, 5, 5, 4, 4, 5, 4, 5, 5, 3, 7, 3, 5, 1, 7, 4, 7, 7, 5, 9, 5, 9, 3, 3, 5, 13, 7, 9, 7, 3, 4, 10, 10, 9, 11

Offset: 1

Author

Zhi-Wei Sun, Jan 24 2014

Keywords

• nonn

Comments

Conjecture: a(n) > 0 for all n > 31.

We have verified this for n up to 60000.

The conjecture implies that there are infinitely many positive integers m with 2^(m-1)*phi(m) - 1 prime.

See A236375 for a list of known numbers m with 2^(m-1)*phi(m) - 1 prime.

Examples

			a(24) = 1 since phi(8)/2 + phi(16)/8 = 3 with 2^(3-1)*phi(3) - 1 = 7 prime.
a(33) = 1 since phi(13)/2 + phi(20)/8 = 7 with 2^(7-1)*phi(7) - 1 = 383 prime.
a(79) = 1 since phi(27)/2 + phi(52)/8 = 9 + 3 = 12 with 2^(12-1)*phi(12) - 1 = 2^(13) - 1 = 8191 prime.
		

Links

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

Crossrefs

Cf. A000010, A000040, A000079, A236375.

Programs

Mathematica

q[n_]:=IntegerQ[n]&&PrimeQ[2^(n-1)*EulerPhi[n]-1]
f[n_,k_]:=EulerPhi[k]/2+EulerPhi[n-k]/8
a[n_]:=Sum[If[q[f[n,k]],1,0],{k,1,n-1}]
Table[a[n],{n,1,100}]