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A048597
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Very round numbers: reduced residue system consists of only primes and 1.
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30
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OFFSET
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1,2
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COMMENTS
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According to Ribenboim, Schatunowsky and Wolfskehl independently showed that 30 is the largest element in the sequence. This gives a lower bound for the maximum of the smallest prime in a, a+d, a+2d, ... taken over all a with 1 < a < d and gcd(a,d) = 1 for d > 30 [see Ribenboim].
It appears that 2, 4, 6, 10, 12 are all the numbers n with the property that every number m in the range n < m < 2n that is coprime to n is also prime. - Ely Golden, Dec 05 2016
Golden's guess is true. See a proof in the links section. - FUNG Cheok Yin, Jun 19 2021
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REFERENCES
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A. H. Beiler, Recreations in the Theory of Numbers, page 91.
H. Rademacher and O. Toeplitz, Von Zahlen und Figuren, Springer Verlag, Berlin, 1933, Zweite Auflage, see last chapter.
H. Rademacher & O. Toeplitz, The Enjoyment of Mathematics, pp. 187-192 Dover Publications, NY 1990.
P. Ribenboim, The little book of big primes, Chapter on primes in arithmetic progression.
J. E. Roberts, Lure of Integers, pp. 179-180 MAA 1992.
J. V. Uspensky and M. A. Heaslet, Elementary Number Theory, McGraw-Hill, NY, 1939, p. 89.
David Wells, The Penguin Dictionary of Curious and Interesting Numbers. Penguin Books, NY, 1986, Revised edition 1987. See p. 111.
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LINKS
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Ross Honsberger, Mathematical Gems, The Two-Year College Mathematics Journal, Vol. 10, No. 3 (Jun., 1979), pp. 195-197 (3 pages).
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EXAMPLE
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The reduced residue systems of these numbers are as follows: {{1, {1}}, {2, {1}}, {3, {1, 2}}, {4, {1, 3}}, {6, {1, 5}}, {8, {1, 3, 5, 7}}, {12, {1, 5, 7, 11}}, {18, {1, 5, 7, 11, 13, 17}}, {24, {1, 5, 7, 11, 13, 17, 19, 23}}, {30, {1, 7, 11, 13, 17, 19, 23, 29}}}.
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MATHEMATICA
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Select[Range[10^3], Function[n, Times @@ Boole@ Map[Or[# == 1, PrimeQ@ #] &, Select[Range@ n, CoprimeQ[#, n] &]] == 1]] (* Michael De Vlieger, Dec 13 2016 *)
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PROG
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CROSSREFS
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The sequences consists of the n with A036997(n)=0.
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KEYWORD
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fini,full,nonn,changed
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AUTHOR
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EXTENSIONS
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Additional comments from Ulrich Schimke (ulrschimke(AT)aol.com), May 29 2001
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STATUS
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approved
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