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A098047 Numbers not in A098006. 3
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

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.

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

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Last modified April 16 04:49 EDT 2021. Contains 343030 sequences. (Running on oeis4.)