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A048597 Very round numbers: reduced residue system consists of only primes and 1. 24
1, 2, 3, 4, 6, 8, 12, 18, 24, 30 (list; graph; refs; listen; history; text; internal format)
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].

For n >= 4, numbers that are divisible by all primes <= sqrt(n). - Jayanta Basu, May 03 2013

REFERENCES

A. H. Beiler, Recreations in the Theory of Numbers, page 91.

H. Bonse, Über eine bekannte Eigenschaft der Zahl 30 und ihre Verallgemeinerung, Archiv der Mathematik und Physik 3 (12) (1907), 292-295.

R. Honsberger, Mathematical Diamonds, MAA, 2003, see p. 79. [Added by N. J. A. Sloane, Jul 05 2009]

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 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

LINKS

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

Bill Taylor, Posting to sci.math, Sep 13 1999

FORMULA

PrimeQ[ {k | GCD[ a[ n ], k ]=1; k= 2, ..., n-1} ] = True for all k.

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}}}

CROSSREFS

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

Sequence in context: A074733 A001461 A173383 * A074964 A017822 A179042

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

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Last modified April 26 15:23 EDT 2015. Contains 257085 sequences.