A237016 a(n) = |{0 < k < n: phi(k)*sigma(n-k) is a square}|, where phi(.) is Euler's totient function and sigma(j) is the sum of all positive divisors of j.

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

Offset: 1

Author

Zhi-Wei Sun, Feb 02 2014

Keywords

• nonn

Comments

Conjecture: (i) a(n) > 0 except for n = 1, 7, 17.

(ii) If n > 5, then phi(k)*sigma(n-k) + 1 is a square for some 0 < k < n.

(iii) If n > 309, then there is a positive integer k < n/2 such that sigma(k)*sigma(n-k) is a square.

See also A236998 for a similar conjecture.

Examples

			a(9) = 1 since phi(8)*sigma(1) = 4*1 = 2^2.
a(16) = 1 since phi(6)*sigma(10) = 2*18 = 6^2.
a(31) = 1 since phi(24)*sigma(7) = 8*8 = 8^2.
a(65) = 1 since phi(19)*sigma(46) = 18*72 = 36^2.
		

Links

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

Crossrefs

Cf. A000010, A000203, A000290, A234246, A236567, A236998.

Programs

Mathematica

sigma[n_]:=DivisorSigma[1,n]
SQ[n_]:=IntegerQ[Sqrt[n]]
p[n_,k_]:=SQ[EulerPhi[k]*sigma[n-k]]
a[n_]:=Sum[If[p[n,k],1,0],{k,1,n-1}]
Table[a[n],{n,1,100}]