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%I #11 Mar 15 2023 12:39:42
%S 5,20,21,22,24,28,31,33,34,36,37,38,43,45,46,48,51,52,55,58,61,67,69,
%T 70,73,79,80,82,87,88,91,97,99,100,104,106,108,112,115,117,118,123,
%U 124,127,130,132,136,138,142,145,147,148,151,152,154,156,157,159,163,166,172
%N Numbers not in A098006.
%C In the Luca-Walsh paper it is shown that this sequence is infinite.
%C 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
%C 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
%H T. D. Noe, <a href="/A098047/b098047.txt">Table of n, a(n) for n=1..1000</a>
%H William D. Banks and Florian Luca, <a href="https://arxiv.org/abs/math/0409231">Noncototients and Nonaliquots</a>, arXiv:math/0409231 [math.NT], 2004.
%H William D. Banks and Florian Luca, <a href="http://dx.doi.org/10.4064%2Fcm103-1-4">Nonaliquots and Robbins numbers</a>, Colloq. Math., Vol. 103, No. 1 (2005), pp. 27-32.
%H Florian Luca and P. G. Walsh, <a href="http://dx.doi.org/10.4064%2Fcm100-1-8">On the number of nonquadratic residues which are not primitive roots</a>, Colloq. Math., Vol. 100, No. 1 (2004), pp. 91-93.
%H T. D. Noe, <a href="http://www.sspectra.com/math/A098006.pdf">Finding primes for which (p-1)/2 - phi(p-1) = k</a>.
%H Neville Robbins, <a href="https://wcnt.files.wordpress.com/2018/02/wcnt-problems-2002.pdf">Problem 002:18</a>, Western Number Theory Problems, 16 & 19 Dec 2002. See p. 8; Florian Luca and Gary Wals, <a href="https://wcnt.files.wordpress.com/2018/02/wcnt-problems-2004.pdf">Solution</a>, Western Number Theory Problems, 17 & 19 Dec 2004. See p. 2.
%t 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]
%Y Cf. A098006.
%K nonn
%O 1,1
%A _Robert G. Wilson v_, Sep 09 2004
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