|
Search: id:A063427
|
|
|
| A063427 |
|
a(n) is the smallest positive integer k such that n*k/(n+k) is an integer. |
|
+0
8
|
|
| 2, 6, 4, 20, 3, 42, 8, 18, 10, 110, 4, 156, 14, 10, 16, 272, 9, 342, 5, 28, 22, 506, 8, 100, 26, 54, 21, 812, 6, 930, 32, 66, 34, 14, 12, 1332, 38, 78, 10, 1640, 7, 1806, 44, 30, 46, 2162, 16, 294, 50, 102, 52, 2756, 27, 66, 8, 114, 58, 3422, 12, 3660, 62, 18, 64, 104
(list; graph; listen)
|
|
|
OFFSET
|
2,1
|
|
|
COMMENT
|
This produces the smallest positive integer value for n*k/(n+k).
Equivalently, smallest c such that 1/n+1/c=1/b has integer solutions.
Largest c is 1/(n(n-1)) since 1/n+1/(n(n-1))=1/(n-1)
Let L(x,y)=x+y be the "basic" linear form. Let Q(x,y)=x^2+x*y+y^2 be the "basic" quadratic form. Let C(x,y)=x^3+y^3+x^2*y+x*y^2+x*y+x^2+y^2+x+y be the "basic" cubic form. Then a(n)=min(x/Q(x,n)=0 mod L(x,n))=min(x/C(x,n)=0 mod L(x,n)). - Benoit Cloitre (benoit7848c(AT)orange.fr), Jan 02 2002
For p=prime, a(p^k) = p^k*(p-1). - Leroy Quet Jan 25 2007
|
|
LINKS
|
Harry J. Smith, Table of n, a(n) for n=2,...,1000
Leroy Quet, Home Page (listed in lieu of email address)
|
|
FORMULA
|
a(n) = n*A063428(n)/(n-A063428(n))
|
|
EXAMPLE
|
a(6)=3 because 6*3/(6+3)=2 is the smallest integer of the form 6*k/(6+k).
a(10) = 10 since 1/10+1/10 = 1/5, 1/10+1/15 = 1/6, 1/10+1/40 = 1/8, 1/10+1/90 = 1/9 and so the first sum provides the value.
|
|
PROGRAM
|
(PARI) { for (n=2, 1000, k=1; while (n*k%(n + k), k++); write("b063427.txt", n, " ", k) ) } [From Harry J. Smith (hjsmithh(AT)sbcglobal.net), Aug 20 2009]
|
|
CROSSREFS
|
Cf. A063428, A127730.
Sequence in context: A033457 A133936 A065350 this_sequence A066092 A100695 A100140
Adjacent sequences: A063424 A063425 A063426 this_sequence A063428 A063429 A063430
|
|
KEYWORD
|
nonn
|
|
AUTHOR
|
Henry Bottomley (se16(AT)btinternet.com), Jul 19 2001
|
|
EXTENSIONS
|
New description from Benoit Cloitre (benoit7848c(AT)orange.fr), Dec 30 2001
Entry revised by N. J. A. Sloane (njas(AT)research.att.com), Feb 13 2007
Definition revised by Franklin T. Adams-Watters (FrankTAW(AT)Netscape.net), Aug 07 2009
|
|
|
Search completed in 0.002 seconds
|
