John H. Mason - cp's OEIS frontend

A299068 Number of pairs of factors of n^2*(n^2-1) which differ by n.

3, 4, 8, 7, 11, 6, 10, 12, 11, 9, 9, 9, 13, 22, 12, 7, 7, 11, 21, 28, 9, 7, 17, 14, 13, 14, 13, 13, 11, 9, 10, 12, 17, 33, 28, 8, 7, 20, 19, 15, 9, 10, 21, 29, 10, 7, 14, 19, 18, 21, 11, 9, 16, 44, 46, 14, 7, 9, 15, 9, 9, 18, 40, 24, 18, 8, 9, 30, 18, 17, 11

Offset: 2

John H Mason, Feb 01 2018

nonn

The question arose when seeking triples of numbers for which the sum of the squares of any two is congruent to 1 modulo the third.
From Robert Israel, Feb 04 2018: (Start)
For n > 7, a(n)>= 7, as there are at least the following pairs:
(1,n+1), (n,2*n), (2*n,3*n), ((n^2-n)/2,(n^2+n)/2), (n^2-n,n^2), (n^2,n^2+n), and (3*n, 4*n) (if n is odd) or (n/2,3*n/2) (if n is even).
If k in A299159 is sufficiently large, then a(12*k-2)=7.  Dickson's conjecture implies there are infinitely many such k, and thus infinitely many n with a(n)=7. (End)

Robert Israel, Table of n, a(n) for n = 2..10000

Cf. A299159.

a:= n-> (s-> add(`if`(i+n in s, 1, 0), i=s))(
        numtheory[divisors](n^2*(n^2-1))):
seq(a(n), n=2..100);  # Alois P. Heinz, Feb 01 2018

Array[With[{d = Divisors[# (# - 1)] &[#^2]}, Count[d + #, ?(MemberQ[d, #] &)]] &, 71, 2] (* _Michael De Vlieger, Feb 01 2018 *)

A121052 Smallest positive integer m for which n^m is congruent to 1 modulo n^2+n-1.

Original entry on oeis.org

1, 4, 5, 9, 14, 40, 20, 35, 44, 108, 65, 60, 45, 90, 119, 135, 60, 30, 189, 209, 46, 100, 63, 299, 145, 700, 100, 135, 390, 928, 99, 84, 522, 280, 629, 605, 56, 1480, 779, 740, 430, 684, 60, 989, 517, 80, 40, 1175, 195, 2548, 240, 252, 715, 424, 81, 1595, 220, 310

Offset: 1

John H. Mason, Aug 09 2006

nonn

The sequence arises as the order of a shuffle of n(n+1) cards in which cards are laid out in an array of n+1 rows of n columns; cards are picked up by column and laid out by rows.
More generally there is a function of two variables, f(r,c) for which f(r,c) is the least integer such that c^f(r,c) is congruent to 1 modulo rc-1. Of interest is the ratio of phi(rc-1)/f(r,c) or in the case of the sequence proposed, phi(n^2+n-1)/m.
I would like to know if there is some direct way to predict these orders, or the ratio of phi(rc-1)/f(r,c). The program provided produces the table f(r,c).

			a(2)=4 because 2^4=16=1 mod 5 but 2^1, 2^2 and 2^3 are not;
a(3)=5 because 3^5=1 mod 11 and 5 is the smallest such.

Alois P. Heinz, Table of n, a(n) for n = 1..20000

TAB:=proc(Rmin,Rmax,Cmin,Cmax) local r,c,T,m,ct,A; T:=array(1..Rmax-Rmin+1,1..Cmax-Cmin+1); for r from Rmin to Rmax do for c from Cmin to Cmax do A:=c;ct:=1;m:=r*c-1; while not A = 1 do A:=A*c mod m;ct:=ct+1; od; T[r-Rmin+1,c-Cmin+1]:=[ct,phi(m)]; od;od; eval(T) end:
# second Maple program:
a:= n-> `if`(n=1, 1, numtheory[order](n, n^2+n-1)):
seq(a(n), n=1..75);  # Alois P. Heinz, Feb 18 2020

f[n_] := If[n == 1, 1, Block[{m = 1, k = n^2 + n - 1}, While[Mod[n^m, k] != 1, m++ ]; m]]; Array[f, 59] (* Robert G. Wilson v *)

print1(1,",");for(n=2,60,q=n^2+n-1;m=1;while(lift(Mod(n,q)^m)!=1,m++);print1(m,",")) \\ Klaus Brockhaus, Aug 09 2006

More terms from Klaus Brockhaus and Robert G. Wilson v, Aug 09 2006

A090767 Numbers of the form 3xyz + 2(xy + yz + zx) + (x + y + z) for x, y, z positive integers.

Original entry on oeis.org

12, 20, 28, 33, 36, 44, 46, 52, 54, 59, 60, 64, 68, 72, 75, 76, 82, 84, 85, 92, 96, 98, 100, 104, 105, 108, 111, 116, 117, 118, 124, 128, 132, 133, 136, 137, 138, 140, 144, 148, 150, 151, 154, 156, 159, 162, 163, 164, 170, 172, 174, 176, 180, 184, 188, 189, 190

Offset: 1

John H. Mason, Feb 02 2004

nonn

This is the set of numbers which count the unit sticks or unit segments needed to construct a three-dimensional cubic lattice made up from unit cubes. This generalizes the two-dimensional version which is A047845 (numbers of the form 2*x*y + x + y for x and y positive integers) and is also the numbers of sticks needed to construct a rectangular lattice of unit squares.

			a(1) = 12 because there are 12 edges to a cube.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A047845.

SeqGen1 := proc(n,N) local a,b,c,F,V,v; # n specifies the search space; N specifies the maximal number to appear in the initial segment of the sequence F := 3*x*y*z + 2*(x*y+y*z+z*x)+x+y+z; V := {}; for a from 1 to n do for b from 1 to n do for c from b to n do v := subs(x=a,y=b,F); if v < N then V := V union {v};fi; od;od; sort(V) end:
# alternative:
N:= 1000: # to get all terms <= N
S:= {seq(seq(seq(3*x*y*z + 2*(x*y+y*z+z*x)+(x+y+z),
z = 1 .. min(y, (-2*x*y+N-x-y)/(3*x*y+2*x+2*y+1))),
y = 1 .. min(x, (N-3*x-1)/(5*x+3))),
x = 1 .. (N-4)/8)}:
sort(convert(S,list)); # Robert Israel, Feb 18 2016

M = 1000;
S = Table[3 x y z + 2(x y + y z + z x) + (x + y + z), {x, 1, (M - 4)/8}, {y, 1, Min[x, (M - 3 x - 1)/(5 x + 3)]}, {z, 1, Min[y, (-2 x y + M - x - y)/(3 x y + 2 x + 2 y + 1)]}] // Flatten // Union (* Jean-François Alcover, Apr 11 2019, after Robert Israel *)

More terms from Ray Chandler, Feb 04 2004

A007989 Slowest increasing sequence with property that a(n) does not divide any term of form a(n+m*a(n)) for any m; this is the slowest increasing "non-difference-divisible" sequence.

Original entry on oeis.org

2, 3, 5, 6, 7, 8, 9, 11, 13, 14, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 37, 41, 42, 43, 44, 47, 49, 53, 54, 55, 58, 59, 61, 65, 66, 67, 68, 69, 70, 71, 73, 75, 76, 77, 79, 83, 85, 87, 88, 89, 91, 95, 96, 97, 98, 101, 103, 107, 109

Offset: 0

John H. Mason

nonn

Sean A. Irvine, Table of n, a(n) for n = 0..1000
J. Mason, Mathematical Abstraction Seen and a Delicate Shift of Attention, For the learning of Mathematics 9 (2) 1989, pp. 2-8.
J. Mason, Difference-Divisible Sequences, Math. Gazette 74 (469) 1990, pp. 223-228.

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User: John H. Mason

A299068 Number of pairs of factors of n^2*(n^2-1) which differ by n.

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A121052 Smallest positive integer m for which n^m is congruent to 1 modulo n^2+n-1.

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A090767 Numbers of the form 3*x*y*z + 2(x*y + y*z + z*x) + (x + y + z) for x, y, z positive integers.

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A007989 Slowest increasing sequence with property that a(n) does not divide any term of form a(n+m*a(n)) for any m; this is the slowest increasing "non-difference-divisible" sequence.

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A090767 Numbers of the form 3xyz + 2(xy + yz + zx) + (x + y + z) for x, y, z positive integers.