A234399 a(n) = |{0 < k < n: 2^k*(2^phi(n-k) - 1) + 1 is prime}|, where phi(.) is Euler's totient function.

0, 1, 2, 2, 3, 3, 2, 3, 5, 4, 4, 5, 3, 6, 5, 3, 6, 8, 4, 5, 5, 6, 4, 6, 7, 4, 5, 6, 4, 3, 4, 9, 5, 3, 8, 5, 4, 3, 8, 8, 3, 8, 6, 7, 7, 8, 8, 9, 4, 5, 8, 9, 7, 6, 10, 11, 4, 6, 6, 8, 8, 10, 4, 4, 7, 4, 12, 8, 6, 4, 9, 7, 4, 6, 10, 9, 8, 7, 7, 7, 5, 4, 10, 5, 6, 7, 9, 15, 7, 8, 10, 7, 4, 8, 6, 10, 3, 3, 10, 11

Offset: 1

Author

Zhi-Wei Sun, Dec 25 2013

Keywords

• nonn

Comments

Conjecture: a(n) > 0 for all n > 1.

See also the conjecture in A234388.

The conjecture is false. a(5962) = 0. - Jason Yuen, Nov 04 2024

Examples

			a(7) = 2 since 2^1*(2^phi(6)-1) + 1 = 2*3 + 1 = 7 and 2^2*(2^phi(5)-1) + 1 = 4*15 + 1 = 61 are both prime.
		

Links

• Zhi-Wei Sun, Table of n, a(n) for n = 1..4000

Crossrefs

Cf. A000010, A000040, A000079, A234309, A234310, A234337, A234344, A234346, A234347, A234359, A234360, A234361, A234388

Programs

Mathematica

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

PARI

a(n) = sum(k=1,n-1,ispseudoprime(2^k*(2^eulerphi(n-k)-1)+1)) \\ Jason Yuen, Nov 04 2024