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A101257 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)).

%I A101257 #11 Feb 16 2025 08:32:55
%S A101257 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 A101257 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 A101257 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 A101257 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 A101257 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 A101257 Antti Karttunen, <a href="/A101257/b101257.txt">Table of n, a(n) for n = 1..65537</a>
%H A101257 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/PointLattice.html">Point Lattice</a>.
%H A101257 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Divisor.html">Divisor</a>.
%e A101257 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 A101257 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 A101257 (PARI)
%o A101257 A033676(n) = if(n<2, 1, my(d=divisors(n)); d[(length(d)+1)\2]); \\ From A033676
%o A101257 A033677(n) = (n/A033676(n));
%o A101257 A101257(n) = (A033677(n)%A033676(n)); \\ _Antti Karttunen_, Sep 23 2018
%Y A101257 Cf. A033676, A033677, A080363.
%K A101257 nonn,look
%O A101257 1,15
%A A101257 Joseph Biberstine (jrbibers(AT)indiana.edu), Dec 17 2004
%E A101257 Definition corrected by _Antti Karttunen_, Sep 23 2018