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

5, 7, 9, 11, 13, 16, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 251, 257, 263, 269, 271

Offset: 1

Author

Labos Elemer, May 07 2003

Keywords

• nonn
• easy

Comments

For primes this means 0 < 2 < p-1 so primes p greater than 3 are members.

Only two composite solutions below 10000000: n = 9 and n = 16.

From Charles R Greathouse IV, Apr 12 2010: (Start)

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.

If n is composite, then the cototient n - phi(n) >= sqrt(n).

For n > 32760, d(n) < sqrt(n)/2.

So all composite solutions are in 1..32760. Checking these (and applying the other inequality), the only composite members are 9 and 16.

Thus the sequence is the primes greater than 3, together with 9 and 16.

(End)

Examples

			n = 9 is a member: 3 divisors, 6 coprimes, 1 (it is 6) unrelated: 6 > 3 > 1;
n = 16 is a member: 5 divisors, 8 coprimes 4 unrelateds ({6, 10, 12, 14}): 8 > 5 > 4.
		

Links

• Michael De Vlieger, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A000005, A000010, A045763, A073757, A051953.

Programs

Mathematica

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 *)
(* Second program: *)
Select[Range@ 272, Function[n, n - (#1 + #2 - 1) < #1 < #2 & @@ {DivisorSigma[0, n], EulerPhi[n]}]] (* Michael De Vlieger, Jul 22 2017 *)

PARI

a(n) = if(n>6,prime(n),[5,7,9,11,13,16][n]) \\ Charles R Greathouse IV, Apr 12 2010

Extensions

Extension, new definition, and edits from Charles R Greathouse IV, Apr 12 2010