A048597 Very round numbers: reduced residue system consists of only primes and 1.

1, 2, 3, 4, 6, 8, 12, 18, 24, 30

Offset: 1

Author

Labos Elemer

Keywords

• fini
• full
• nonn

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

Examples

			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}}}.
		

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]
• Yuto Kaneko and Hirofumi Nakai, A Generalization of Schatunowsky's Theorem, Amer. Math. Monthly (2025). See p. 1.
• 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.

Crossrefs

The sequences consists of the n with A036997(n)=0.

Programs

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 *)

PARI

is(n)=forcomposite(k=2,n-1,if(gcd(n,k)==1, return(0))); 1 \\ Charles R Greathouse IV, Apr 28 2015

Extensions

Additional comments from Ulrich Schimke (ulrschimke(AT)aol.com), May 29 2001