A236567 a(n) = |{0 < k < n - 2: k + phi(n-k)/2 is a square}|, where phi(.) is Euler's totient function.

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

Offset: 1

Author

Zhi-Wei Sun, Feb 02 2014

Keywords

• nonn

Comments

Conjecture: (i) a(n) > 0 for all n > 5.

(ii) For n > 31, there is a positive integer k < n - 2 with phi(k) + phi(n-k)/2 a square. If n > 70 is not equal to 150, then phi(k) + phi(n-k) is a square for some 0 < k < n.

(iii) If n > 5, then phi(k) + phi(n-k)/2 is a triangular number for some 0 < k < n - 2. For each n = 20, 21, ... there is a positive integer k < n with (phi(k) + phi(n-k))/2 a triangular number.

Examples

			a(8) = 1 since 1 + phi(7)/2 = 1 + 3 = 2^2.
a(11) = 1 since 8 + phi(3)/2 = 8 + 1 = 3^2.
a(78) = 1 since 40 + phi(38)/2 = 40 + 9 = 7^2.
		

Links

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

Crossrefs

Cf. A000010, A000290, A234246.

Programs

Mathematica

SQ[n_]:=IntegerQ[Sqrt[n]]
p[n_,k_]:=SQ[k+EulerPhi[n-k]/2]
a[n_]:=Sum[If[p[n,k],1,0],{k,1,n-3}]
Table[a[n],{n,1,100}]