OFFSET
1,2
COMMENTS
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
REFERENCES
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.
LINKS
H. Bonse, Über eine bekannte Eigenshaft der Zahl 30 und ihre Verallgemeinerung, Archiv d. Math. u. Physik (3) vol. 12 (1907) 292-295.
Ross Honsberger, Mathematical Gems, The Two-Year College Mathematics Journal, Vol. 10, No. 3 (Jun., 1979), pp. 195-197 (3 pages).
Ross Honsberger, Two distinguished integers, in Mathematical Diamonds, MAA, 2003, see p. 79. [Added by N. J. A. Sloane, Jul 05 2009]
Bill Taylor, Posting to sci.math, Sep 13 1999 [Broken link]
Fung Cheok Yin, A property of the set "2, 4, 6, 10, 12", Dec 24 2020.
EXAMPLE
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}}}.
MATHEMATICA
Select[Range[10^3], Function[n, Times @@ Boole@ Map[Or[# == 1, PrimeQ@ #] &, Select[Range@ n, CoprimeQ[#, n] &]] == 1]] (* Michael De Vlieger, Dec 13 2016 *)
PROG
(PARI) is(n)=forcomposite(k=2, n-1, if(gcd(n, k)==1, return(0))); 1 \\ Charles R Greathouse IV, Apr 28 2015
CROSSREFS
KEYWORD
fini,full,nonn
AUTHOR
EXTENSIONS
Additional comments from Ulrich Schimke (ulrschimke(AT)aol.com), May 29 2001
STATUS
approved