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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 February 28 15:42 EST 2015. Contains 255088 sequences.