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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

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Last modified December 11 12:57 EST 2009. Contains 170656 sequences.


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