A098047 Numbers not in A098006.

5, 20, 21, 22, 24, 28, 31, 33, 34, 36, 37, 38, 43, 45, 46, 48, 51, 52, 55, 58, 61, 67, 69, 70, 73, 79, 80, 82, 87, 88, 91, 97, 99, 100, 104, 106, 108, 112, 115, 117, 118, 123, 124, 127, 130, 132, 136, 138, 142, 145, 147, 148, 151, 152, 154, 156, 157, 159, 163, 166, 172

Offset: 1

Author

Robert G. Wilson v, Sep 09 2004

Keywords

• nonn

Comments

In the Luca-Walsh paper it is shown that this sequence is infinite.

It can be shown that if a number k > 8, k not a power of 2, is in A098006, then k first appears for a prime p <= 1+k^2. For example, 26 first appears as A098006(123). The 123rd prime is 677, which equals 1+26^2. When this worst-case behavior occurs, then k/2 is a prime in A052291 and the corresponding 1+k^2 is in A052292. - T. D. Noe, Nov 13 2007

Banks and Luca (2004, 2005) called these numbers Robbins numbers. They proved that the lower asymptotic density of this sequence is > 1/3. - Amiram Eldar, Feb 13 2021

Links

• T. D. Noe, Table of n, a(n) for n=1..1000
• William D. Banks and Florian Luca, Noncototients and Nonaliquots, arXiv:math/0409231 [math.NT], 2004.
• William D. Banks and Florian Luca, Nonaliquots and Robbins numbers, Colloq. Math., Vol. 103, No. 1 (2005), pp. 27-32.
• Florian Luca and P. G. Walsh, On the number of nonquadratic residues which are not primitive roots, Colloq. Math., Vol. 100, No. 1 (2004), pp. 91-93.
• T. D. Noe, Finding primes for which (p-1)/2 - phi(p-1) = k.
• Neville Robbins, Problem 002:18, Western Number Theory Problems, 16 & 19 Dec 2002. See p. 8; Florian Luca and Gary Wals, Solution, Western Number Theory Problems, 17 & 19 Dec 2004. See p. 2.

Crossrefs

Cf. A098006.

Programs

Mathematica

t = Table[0, {200}]; Do[p = Prime[n]; a = (p - 1)/2 - EulerPhi[p - 1]; If[p < 201, t[[a]]++ ], {n, 2, 10^7}]; u = Table[ If[ t[[n]] != 0, n, 0], {n, 1, 200}]; Complement[ Range[200], u]