This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A063427 #39 Dec 31 2024 14:04:03
%S A063427 2,6,4,20,3,42,8,18,10,110,4,156,14,10,16,272,9,342,5,28,22,506,8,100,
%T A063427 26,54,21,812,6,930,32,66,34,14,12,1332,38,78,10,1640,7,1806,44,30,46,
%U A063427 2162,16,294,50,102,52,2756,27,66,8,114,58,3422,12,3660,62,18,64,104
%N A063427 a(n) is the smallest positive integer k such that n*k/(n+k) is an integer.
%C A063427 This produces the smallest positive integer value for n*k/(n+k).
%C A063427 Equivalently, smallest c such that 1/n + 1/c = 1/b has integer solutions.
%C A063427 Largest c is 1/(n(n-1)) since 1/n + 1/(n(n-1)) = 1/(n-1).
%C A063427 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_, Jan 02 2002
%C A063427 For p=prime, a(p^k) = p^k*(p-1). - _Leroy Quet_, Jan 25 2007
%C A063427 a(n) = n*(d(i)-d(i-1))/d(i-1), where d(i) is the i-th divisor of n that minimizes (d(i)-d(i-1))/d(i-1) with i>=2. In general, let f(n) be an integer function, then n*f(n)/(n+f(n))=c, c positive integer, has a solution only if f(n) >= n*(d(i)-d(i-1))/d(i-1). - _Ctibor O. Zizka_, Sep 17 2015
%H A063427 Harry J. Smith, <a href="/A063427/b063427.txt">Table of n, a(n) for n = 2..1000</a>
%F A063427 a(n) = n*A063428(n)/(n-A063428(n)).
%e A063427 a(6) = 3 because 6*3/(6+3)=2 is the smallest integer of the form 6*k/(6+k).
%e A063427 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.
%t A063427 Table[k=1;While[!IntegerQ[(k n)/(k+n)],k++];k,{n,2,70}] (* _Harvey P. Dale_, Jun 24 2011 *)
%o A063427 (PARI) a(n) = { my(k=1); while (n*k%(n + k), k++); k } \\ _Harry J. Smith_, Aug 20 2009
%Y A063427 Cf. A063428, A127730.
%K A063427 nonn
%O A063427 2,1
%A A063427 _Henry Bottomley_, Jul 19 2001
%E A063427 New description from _Benoit Cloitre_, Dec 30 2001
%E A063427 Entry revised by _N. J. A. Sloane_, Feb 13 2007
%E A063427 Definition revised by _Franklin T. Adams-Watters_, Aug 07 2009