

A048597


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


29




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


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. 187192 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. 179180 MAA 1992.
J. V. Uspensky and M. A. Heaslet, Elementary Number Theory, McGrawHill, 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) 292295.
Ross Honsberger, Mathematical Gems, The TwoYear College Mathematics Journal, Vol. 10, No. 3 (Jun., 1979), pp. 195197 (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]


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, n1, 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: A074733 A001461 A173383 * 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



