A101257 - OEIS

%I #10 Sep 23 2018 20:57:43

%S 0,0,0,0,0,1,0,0,0,1,0,1,0,1,2,0,0,0,0,1,1,1,0,2,0,1,0,3,0,1,0,0,2,1,

%T 2,0,0,1,1,3,0,1,0,3,4,1,0,2,0,0,2,1,0,3,1,1,1,1,0,4,0,1,2,0,3,5,0,1,

%U 2,3,0,1,0,1,0,3,4,1,0,2,0,1,0,5,2,1,2,3,0,1,6,3,1,1,4,4,0,0,2,0,0,5,0,5,1

%N Remainder when the least divisor of n greater than or equal to the square root of n (A033677(n)) is divided by the greatest divisor of n less than or equal to the square root of n (A033676(n)).

%C Given n points, sort them into the most-square rectangular point lattice possible. Now sort the points into square point lattices of dimension equal to the lesser dimension of the earlier rectangle. a(n) is the number of points left over. a(n) is trivially 0 for prime numbers n (the most-square and only rectangular point lattice on a prime number of points is a linear point lattice). a(n) != 0 iff n is a member of A080363.

%H Antti Karttunen, <a href="/A101257/b101257.txt">Table of n, a(n) for n = 1..65537</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/PointLattice.html">Point Lattice</a>.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Divisor.html">Divisor</a>.

%e a(6)=1 because the least divisor of 6 greater than sqrt(6) is 3, the greater divisor of 6 less than sqrt(6) is 2 and 3 mod 2 = 1

%t num[n_] := If[OddQ[DivisorSigma[0, n]], Sqrt[n], Divisors[n][[DivisorSigma[0, n]/2 + 1]]] den[n_] := If[OddQ[DivisorSigma[0, n]], Sqrt[n], Divisors[n][[DivisorSigma[0, n]/2]]] Table[Mod[num[n], den[n]], {n, 1, 128}]

%o (PARI)

%o A033676(n) = if(n<2, 1, my(d=divisors(n)); d[(length(d)+1)\2]); \\ From A033676

%o A033677(n) = (n/A033676(n));

%o A101257(n) = (A033677(n)%A033676(n)); \\ _Antti Karttunen_, Sep 23 2018

%Y Cf. A033676, A033677, A080363.

%K nonn,look

%O 1,15

%A Joseph Biberstine (jrbibers(AT)indiana.edu), Dec 17 2004

%E Definition corrected by _Antti Karttunen_, Sep 23 2018