



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
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OFFSET

1,1


COMMENTS

In the LucaWalsh 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 worstcase 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. 2732.
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. 9193.
T. D. Noe, Finding primes for which (p1)/2  phi(p1) = 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.


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]


CROSSREFS

Cf. A098006.
Sequence in context: A227109 A243800 A335555 * A231276 A101728 A053240
Adjacent sequences: A098044 A098045 A098046 * A098048 A098049 A098050


KEYWORD

nonn


AUTHOR

Robert G. Wilson v, Sep 09 2004


STATUS

approved



