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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)).
1
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, 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, 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
OFFSET
1,15
COMMENTS
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.
LINKS
Eric Weisstein's World of Mathematics, Point Lattice.
Eric Weisstein's World of Mathematics, Divisor.
EXAMPLE
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
MATHEMATICA
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}]
PROG
(PARI)
A033676(n) = if(n<2, 1, my(d=divisors(n)); d[(length(d)+1)\2]); \\ From A033676
A033677(n) = (n/A033676(n));
A101257(n) = (A033677(n)%A033676(n)); \\ Antti Karttunen, Sep 23 2018
CROSSREFS
KEYWORD
nonn,look
AUTHOR
Joseph Biberstine (jrbibers(AT)indiana.edu), Dec 17 2004
EXTENSIONS
Definition corrected by Antti Karttunen, Sep 23 2018
STATUS
approved