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A083249 Numbers n with A045763(n) = n + 1 - d(n) - phi(n) < d(n) < phi(n).

%I A083249 #17 Jul 23 2017 03:18:46
%S A083249 5,7,9,11,13,16,17,19,23,29,31,37,41,43,47,53,59,61,67,71,73,79,83,89,
%T A083249 97,101,103,107,109,113,127,131,137,139,149,151,157,163,167,173,179,
%U A083249 181,191,193,197,199,211,223,227,229,233,239,241,251,257,263,269,271
%N A083249 Numbers n with A045763(n) = n + 1 - d(n) - phi(n) < d(n) < phi(n).
%C A083249 For primes this means 0 < 2 < p-1 so primes p greater than 3 are members.
%C A083249 Only two composite solutions below 10000000: n = 9 and n = 16.
%C A083249 From _Charles R Greathouse IV_, Apr 12 2010: (Start)
%C A083249 d(n) < phi(n) is true for all n > 30 (see A020490), so the main condition is n + 1 - d(n) - phi(n) < d(n). Rewrite this as n - phi(n) < 2d(n) - 1.
%C A083249 If n is composite, then the cototient n - phi(n) >= sqrt(n).
%C A083249 For n > 32760, d(n) < sqrt(n)/2.
%C A083249 So all composite solutions are in 1..32760. Checking these (and applying the other inequality), the only composite members are 9 and 16.
%C A083249 Thus the sequence is the primes greater than 3, together with 9 and 16.
%C A083249 (End)
%H A083249 Michael De Vlieger, <a href="/A083249/b083249.txt">Table of n, a(n) for n = 1..10000</a>
%e A083249 n = 9 is a member: 3 divisors, 6 coprimes, 1 (it is 6) unrelated: 6 > 3 > 1;
%e A083249 n = 16 is a member: 5 divisors, 8 coprimes 4 unrelateds ({6, 10, 12, 14}): 8 > 5 > 4.
%t A083249 Do[r=EulerPhi[n]; d=DivisorSigma[0, n]; u=n-r-d+1; If[Greater[r, d]&&Greater[d, u]&&!PrimeQ[n], Print[n, {d, r, u}]], {n, 1, 1000}] (* for composite solutions *) (* corrected by _Charles R Greathouse IV_, Apr 12 2010 *)
%t A083249 (* Second program: *)
%t A083249 Select[Range@ 272, Function[n, n - (#1 + #2 - 1) < #1 < #2 & @@ {DivisorSigma[0, n], EulerPhi[n]}]] (* _Michael De Vlieger_, Jul 22 2017 *)
%o A083249 (PARI) a(n) = if(n>6,prime(n),[5,7,9,11,13,16][n]) \\ _Charles R Greathouse IV_, Apr 12 2010
%Y A083249 Cf. A000005, A000010, A045763, A073757, A051953.
%K A083249 nonn,easy
%O A083249 1,1
%A A083249 _Labos Elemer_, May 07 2003
%E A083249 Extension, new definition, and edits from _Charles R Greathouse IV_, Apr 12 2010