A237016 - OEIS

%I #11 Feb 02 2014 10:48:28

%S 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,

%T 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,

%U 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

%N 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.

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

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

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

%C See also A236998 for a similar conjecture.

%H Zhi-Wei Sun, <a href="/A237016/b237016.txt">Table of n, a(n) for n = 1..10000</a>

%e a(9) = 1 since phi(8)*sigma(1) = 4*1 = 2^2.

%e a(16) = 1 since phi(6)*sigma(10) = 2*18 = 6^2.

%e a(31) = 1 since phi(24)*sigma(7) = 8*8 = 8^2.

%e a(65) = 1 since phi(19)*sigma(46) = 18*72 = 36^2.

%t sigma[n_]:=DivisorSigma[1,n]

%t SQ[n_]:=IntegerQ[Sqrt[n]]

%t p[n_,k_]:=SQ[EulerPhi[k]*sigma[n-k]]

%t a[n_]:=Sum[If[p[n,k],1,0],{k,1,n-1}]

%t Table[a[n],{n,1,100}]

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

%K nonn

%O 1,11

%A _Zhi-Wei Sun_, Feb 02 2014