cp's OEIS Frontend

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.

%I A237016 #11 Feb 02 2014 10:48:28
%S A237016 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 A237016 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 A237016 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 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.
%C A237016 Conjecture: (i) a(n) > 0 except for n = 1, 7, 17.
%C A237016 (ii) If n > 5, then phi(k)*sigma(n-k) + 1 is a square for some 0 < k < n.
%C A237016 (iii) If n > 309, then there is a positive integer k < n/2 such that sigma(k)*sigma(n-k) is a square.
%C A237016 See also A236998 for a similar conjecture.
%H A237016 Zhi-Wei Sun, <a href="/A237016/b237016.txt">Table of n, a(n) for n = 1..10000</a>
%e A237016 a(9) = 1 since phi(8)*sigma(1) = 4*1 = 2^2.
%e A237016 a(16) = 1 since phi(6)*sigma(10) = 2*18 = 6^2.
%e A237016 a(31) = 1 since phi(24)*sigma(7) = 8*8 = 8^2.
%e A237016 a(65) = 1 since phi(19)*sigma(46) = 18*72 = 36^2.
%t A237016 sigma[n_]:=DivisorSigma[1,n]
%t A237016 SQ[n_]:=IntegerQ[Sqrt[n]]
%t A237016 p[n_,k_]:=SQ[EulerPhi[k]*sigma[n-k]]
%t A237016 a[n_]:=Sum[If[p[n,k],1,0],{k,1,n-1}]
%t A237016 Table[a[n],{n,1,100}]
%Y A237016 Cf. A000010, A000203, A000290, A234246, A236567, A236998.
%K A237016 nonn
%O A237016 1,11
%A A237016 _Zhi-Wei Sun_, Feb 02 2014