A052157 - OEIS

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A052157

Least positive integer r such that there exists an integer s, 0 <= s < r gcd(r-i, s-j) > 1 for all integers i, j with 0 <= i, j < n.

2, 21, 1310, 10199373 (list; graph; refs; listen; history; text; internal format)

	OFFSET	`1,1`
	LINKS	`Table of n, a(n) for n=1..4.` `S. Laishram, F. Luca, Rectangles Of Nonvisible Lattice Points, J. Int. Seq. 18 (2015) 15.10.8`
	FORMULA	`a_n < e^{(1+o(1)) 2 n^2 log n}.`
	EXAMPLE	`a(1) = 2 because we can choose r = 2, s = 0; a(2) = 21 because we can choose r = 21, s = 15; a(3) = 1310 because we can choose r = 1310, s = 1276.` `a(4) = 10199373: r = 10199373, s = 7247646. - William Rex Marshall, Nov 23 2009`
	PROG	`(PARI) is(r, n)=n--; my(s=[Mod(0, 1)], f, v, p); for(i=0, n, f=factor(r-i)[, 1]; for(j=0, n, v=List(); for(k=1, #f, p=f[k]; for(k=1, #s, if(s[k].mod%p, listput(v, chinese(Mod(j, p), s[k])), if(Mod(s[k], p)==j, listput(v, s[k]))))); s=select(m->lift(m)<r, Set(v)); if(#s==0, return(0)))); s[1]` `a(n)=my(r=n); while(!is(r++, n), ); r \\ Charles R Greathouse IV, Nov 05 2013`
	CROSSREFS	`Sequence in context: A015193 A022488 A264066 * A050204 A022470 A080815` `Adjacent sequences: A052154 A052155 A052156 * A052158 A052159 A052160`
	KEYWORD	`nonn,nice,more`
	AUTHOR	`Jeffrey Shallit, Jan 25 2000`
	EXTENSIONS	`By brute force search I know that a(4) > 410000. And also I know by constructing the pair (r, s) = (477742707, 172379781) that a(4) <= 477742707.` `a(4) > 1475000. - Jud McCranie, Jan 26 2000` `a(4) from William Rex Marshall, Nov 23 2009`
	STATUS	`approved`

Last modified April 13 09:12 EDT 2021. Contains 342935 sequences. (Running on oeis4.)