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

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

Offset

1,2

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.

Links

Table of n, a(n) for n=1..10.

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

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

Sequence in context: A107368 A074733 A001461 * A332839 A319054 A074964

Adjacent sequences: A048594 A048595 A048596 * A048598 A048599 A048600

Keyword

fini,full,nonn

Author

Labos Elemer

Extensions

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

Status

approved