Lorenz H. Menke, Jr. - cp's OEIS frontend

A358399 a(n) is the number of reducible monic quartic polynomials (x^4 + rx^3 + sx^2 + t*x + u) with integer coefficients bounded by naïve height n (abs(r), abs(s), abs(t), abs(u) <= n).

47, 271, 810, 1849, 3395, 5832, 8915, 13242, 18465, 25267, 32874, 43023, 53662, 66957, 81770, 99374, 117564, 140303, 163048, 190757, 219702, 252465, 285820, 326853, 366732

Offset: 1

Lorenz H. Menke, Jr., Nov 13 2022

Cf. A358398, A358400, A067274.

{ a(n) = \\ A358399
    my( ct = 0 );
    for (c1 = -n, n,
    for (c2 = -n, n,
    for (c3 = -n, n,
    for (c4 = -n, n,
        if ( ! polisirreducible( Pol([1,c1,c2,c3,c4]) ), ct += 1 );
    ); ); ); );
    return( ct );
}
vector(12, n, a(n) )
\\ Joerg Arndt, Dec 05 2022

A358400 a(n) is the number of reducible monic quintic polynomials (x^5 + rx^4 + sx^3 + tx^2 + ux + v) with integer coefficients bounded by naïve height n (abs(r), abs(s), abs(t), abs(u), abs(v) <= n).

Original entry on oeis.org

139, 1313, 5359, 15365, 34229, 68385, 120421, 200839, 312057, 468827, 669591, 943175, 1274089, 1701441, 2216841, 2856379, 3594651, 4510437, 5541135, 6788823, 8195941, 9845089, 11670925, 13842429, 16191555

Offset: 1

Lorenz H. Menke, Jr., Nov 13 2022

Cf. A358398, A358399, A067274.

{ a(n) = \\ A358400
    my( ct = 0 );
    for (c1 = -n, n,
    for (c2 = -n, n,
    for (c3 = -n, n,
    for (c4 = -n, n,
    for (c5 = -n, n,
        if ( ! polisirreducible( Pol([1,c1,c2,c3,c4,c5]) ), ct += 1 );
    ); ); ); ); );
    return( ct );
}
vector(7, n, a(n) )
\\ Joerg Arndt, Dec 05 2022

A358398 a(n) is the number of reducible monic cubic polynomials x^3 + rx^2 + sx + t with integer coefficients bounded by naïve height n (abs(r), abs(s), abs(t) <= n).

Original entry on oeis.org

15, 53, 117, 215, 329, 493, 657, 877, 1103, 1383, 1643, 2017, 2325, 2721, 3131, 3601, 4009, 4575, 5031, 5647, 6221, 6849, 7409, 8211, 8849, 9593, 10335, 11199, 11899, 12915, 13671, 14655, 15559, 16535, 17473, 18711, 19619, 20711, 21787, 23099, 24095, 25507, 26571, 27931, 29259

Offset: 1

Lorenz H. Menke, Jr., Nov 13 2022

nonn

Artūras Dubickas, On the number of reducible polynomials of bounded naive height, manuscripta math. 144, 439-456 (2014).
Phyllis Lefton, On the Galois groups of cubics and trinomials, Bull. Amer. Math. Soc., vol. 82 (1976), pp. 754-756.
Phyllis Lefton, On the Galois groups of cubics and trinomials, Acta Arithmetica (1979) Volume: 35, Issue: 3, page 239-246.

Cf. A067274.

{ a(n) =
    my( ct = 0 );
    for (c1 = -n, n,
    for (c2 = -n, n,
    for (c3 = -n, n,
        if ( ! polisirreducible( Pol([1,c1,c2,c3]) ), ct += 1 );
    ); ); );
    return( ct );
}
vector(12, n, a(n) ) \\ Joerg Arndt, Dec 12 2022

Dubickas (2014) shows that a(n) ~ 2(1+(2/3)Pi^2)n^2 = 15.1598... n^2 for large n.

a(26)-a(45) from Hugo Pfoertner, Nov 27 2022

A281914 Number of ordered integer pairs (b,c), with -n<=b<=n, -n<=c<=n, such that both roots of 3x^2+bx+c = 0 are rational and b and are not both multiples of 3.

Original entry on oeis.org

2, 8, 8, 16, 24, 24, 34, 46, 46, 60, 72, 74, 86, 100, 104, 122, 132, 132, 142, 164, 168, 182, 192, 200, 214, 228, 228, 250, 260, 268, 278, 300, 304, 318, 336, 340, 350, 364, 368, 398, 408, 416, 426, 448, 452, 466, 476, 488, 502, 524

Offset: 1

Lorenz H. Menke, Jr., Feb 02 2017

nonn

We are not counting the cases where there is a possible overall factor of 3. When there is an overall factor of 3 we get the sequence A067274. These results have been proved and will appear in an upcoming paper.

			The four quadratics for a(2)=8 and their roots are as follows:
3*x^2 + 2*x + 0 = x(2 + 3*x);         x =  0, x = -2/3.
3*x^2 + 2*x - 1 = (1 + x)(- 1 + 3*x); x = -1, x = +1/3.
3*x^2 + 1*x + 0 = x(1 + 3*x);         x =  0, x = -1/3.
3*x^2 + 1*x - 2 = (1 + x)(- 2 + 3*x); x = -1, x = +2/3.
3*x^2 - 1*x + 0 = x(- 1 + 3*x);       x =  0, x = +1/3.
3*x^2 - 1*x - 2 = (- 1 + x)(2 + 3*x); x = +1, x = -2/3.
3*x^2 - 2*x + 0 = x(- 2 + 3*x);       x =  0, x = +2/3.
3*x^2 - 2*x - 1 = (- 1 + x)(1 + 3*x); x = +1, x = -1/3.
There is one case where there is an overall factor of 3 which is counted in series A067274.

Cf. A067274.

a[n_] :=
2 (2 + Floor[(n + 1)/3] + Floor[(n - 1)/3] + Floor[(n + 2)/3] +
     Floor[(n - 2)/3]) +
  2 (KroneckerDelta[4, If[n == 4, 4, 0]] -
     KroneckerDelta[8, If[n == 8, 8, 0]] -
     KroneckerDelta[9, If[n == 9, 9, 0]] -
     KroneckerDelta[10, If[n == 10, 10, 0]] -
     KroneckerDelta[12, If[n == 12, 12, 0]]) +
  If[n >= 4,
   2 (-4 - 2 n - 2 Floor[n/2] + Floor[2 (n + 1)/3] +
      2 Sum[Length[Divisors[k]], {k, n}] -
      2 Sum[Length[Divisors[k]], {k, Floor[n/3]}]), 0];
(* The KroneckerDelta is a special case correction term. *)
a[1] = 2; (* Extends the a[n] series by direct count. *)

A281913 Number of ordered integer pairs (b,c), with -n<=b<=n, -n<=c<=n, such that both roots of 2x^2+bx+c=0 are rational and b and c are not both even.

Original entry on oeis.org

4, 4, 12, 12, 22, 24, 36, 36, 50, 54, 64, 68, 78, 82, 100, 100, 110, 118, 128, 132, 150, 154, 164, 168, 182, 186, 204, 208, 218, 230, 240, 240, 258, 262, 280, 288, 298, 302, 320, 324, 334, 346, 356, 360, 386, 390, 400, 404, 418, 426

Offset: 1

Lorenz H. Menke, Jr., Feb 02 2017

nonn

We are not counting the cases where there is a possible overall factor of 2. When there is an overall factor of 2 we obtain the sequence A067274. These results have been proved and will appear in an upcoming paper.

			The four quadratics for a(2)=4 and their roots are as follows:
2*x^2 + 1*x + 0 = x(1 + 2*x);         x =  0, x = -1/2.
2*x^2 + 1*x - 1 = (1 + x)(- 1 + 2*x); x = -1, x = +1/2.
2*x^2 - 1*x + 0 = x(- 1 + 2*x);       x =  0, x = +1/2.
2*x^2 - 1*x - 1 = (- 1+ x)(1 + 2*x);  x = +1, x = -1/2.
There are nine cases where there is an overall factor of 2 which are counted in series A067274.

Cf. A067274.

a[n_] := If[n >= 3,
   2 (-2 - 2 n + Floor[(n + 1)/2] +
      2 Sum[Length[Divisors[k]], {k, n}] -
      2 Sum[Length[Divisors[k]], {k, Floor[n/2]}]), 0] +
  4 Floor[(n + 1)/2] - 2 KroneckerDelta[6, If[n == 6, 6, 0]];
(* The KroneckerDelta is a special case correction term. *)
a[1] = 4; (* Extends the a[n] series by direct count. *)

A281376 Total number of counts where floor(N/k) < floor((N+k)/n) for k = {1, 2, ..., n-1} and N >= n.

Original entry on oeis.org

0, 0, 0, 1, 3, 6, 11, 17, 25, 35, 47, 60, 77, 95, 115, 138, 164, 191, 222, 254, 290, 329, 370, 412, 460, 510, 562, 617, 676, 736, 802, 869, 940, 1014, 1090, 1169, 1255, 1342, 1431, 1523, 1621, 1720, 1825, 1931, 2041, 2156, 2273, 2391, 2517, 2645, 2777

Offset: 1

Lorenz H. Menke, Jr., Jan 20 2017

nonn

			For n = 5, we have counted the cases where floor(N/k) < floor((N+k)/5), k = {1,2,3,4} then a(5) = 3, since this is true only for k = 4 and N = 6, k = 4 and N = 7, and k = 4 and N = 11.

Jon E. Schoenfield, Table of n, a(n) for n = 1..10000 (terms 1..200 from Lorenz H. Menke, Jr.)

A281376 := proc(n)
    local a,k,N;
    a := 0 ;
    for k from 1 to n-1 do
        for N from n do
            if floor(N/k) < floor((N+k)/n) then
                a := a +1 ;
            elif N >= (k+2*n)*k/(n-k) then
                break;
            end if;
        end do:
    end do:
    a ;
end proc:
seq(A281376(n),n=1..10) ; # R. J. Mathar, Feb 03 2017

a[n_] :=
  Block[{lhs, rhs, count},
   count = 0;
   Do[lhs = Floor[H1/k];
    rhs = Floor[(H1 + k)/n];
    If[lhs < rhs, count++], {k, 1, n - 1}, {H1,
     n, (n^2 - 3 n + 1) + 10}]; (* the 10 is simply guard counts *)
   Return[count]];
a281376[n_] :=
Sum[Floor[j/d], {d, Ceiling[(n - 3)/3]}, {j, n - (2 d + 1)}]
(* Hartmut F. W. Hoft, Jan 25 2017; based on the first formula above *)

a(n) = sum(d = 1, ceil((n-3)/3), sum(j = 1, n-(2*d+1), j\d)); \\ Michel Marcus, Jan 29 2017

a(n) = Sum_{d=1..ceiling((n-3)/3)} Sum_{j=1..n-(2*d+1)} floor(j/d). - Jon E. Schoenfield, Jan 23 2017

a(n) = Sum_{d=1..ceiling(n/3)-1} ((j+1)*(j*d/2 + n mod d)), where j = floor(n/d) - 3. - Jon E. Schoenfield, Jan 24 2017

A270544 Number of ordered pairs (i,j) with |i|, |j| <= n, |i * j| <= n, and i odd.

Original entry on oeis.org

0, 6, 10, 20, 24, 34, 42, 52, 56, 70, 78, 88, 96, 106, 114, 132, 136, 146, 158, 168, 176, 194, 202, 212, 220, 234, 242, 260, 268, 278, 294, 304, 308, 326, 334, 352, 364, 374, 382, 400, 408, 418, 434, 444, 452, 478, 486, 496, 504, 518, 530, 548, 556, 566, 582, 600

Offset: 0

Lorenz H. Menke, Jr., Mar 18 2016

nonn

			a(0) = 0 from (i,j) = ().
a(1) = 6 from (i,j) = (-1,+-1), (1,+-1), (+-1,0).
a(2) = 10: (-1,+-2), (-1,+-1), (+-1,0), (1, +-2), (1,+-1).

The corresponding sequence where i is even is A270543.

a[n_] := 2 Floor[(n+1)/2] + 4 Sum[Floor[n/(2k+1)], {k,0,Floor[(n+1)/2]-1}]

a(n) = {my(nb = 0); for(i=-n, n, if ((i % 2), for(j=-n, n, if (abs(i*j) <= n, nb++);););); nb;} \\ Michel Marcus, Apr 10 2016

a(n) = 2*floor((n+1)/2) + 4*Sum_{k=0..floor((n+1)/2)-1}floor(n/(2k+1)).

a(n) = A226355(n) - A270543(n).

A270543 Number of ordered pairs (i,j) of integers with |i|, |j| <= n, |i * j| <= n, and i even.

Original entry on oeis.org

1, 3, 11, 13, 25, 27, 39, 41, 57, 59, 71, 73, 93, 95, 107, 109, 129, 131, 147, 149, 169, 171, 183, 185, 213, 215, 227, 229, 249, 251, 271, 273, 297, 299, 311, 313, 341, 343, 355, 357, 385, 387, 407, 409, 429, 431, 443, 445, 481, 483, 499, 501, 521, 523, 543, 545

Offset: 0

Lorenz H. Menke, Jr., Mar 18 2016

nonn

			a(0) = 1 from (i,j) = (0,0).
a(1) = 3 from (i,j) = (0,0), (0,1), (0,-1).
a(2) = 11: (0,0), (0,+-1), (0,+-2), (+-2, 0), (+-2,+-1).

The corresponding sequence where i is odd is A270544.

Cf. A226355.

a[n_]:=1+2n+2Floor[n/2]+4Sum[Floor[n/(2k)],{k,1,Floor[n/2]}]

a(n) = {my(nb = 0); for (i=-n, n, if ((i % 2) == 0, for(j=-n, n, if (abs(i*j) <= n, nb++);););); nb;}

a(n) = 1 + 2*n + 2*floor(n/2) + 4*Sum_{k=1..floor(n/2)}floor(n/(2k)).

a(n) = A226355(n) - A270544(n).

a(44) = 429 corrected by Georg Fischer, Sep 13 2023

A270383 Number of ordered pairs (i,j) with i >= j, |i|, |j| <= n, and |i * j| <= n.

Original entry on oeis.org

1, 6, 12, 18, 27, 33, 43, 49, 59, 68, 78, 84, 98, 104, 114, 124, 137, 143, 157, 163, 177, 187, 197, 203, 221, 230, 240, 250, 264, 270, 288, 294, 308, 318, 328, 338, 359, 365, 375, 385, 403, 409, 427, 433, 447, 461, 471, 477, 499, 508, 522

Offset: 0

Lorenz H. Menke, Jr., Mar 15 2016

nonn

			For n = 2 the a(2) = 12 pairs are (2,1), (2,0), (2,-1), (1,1), (1,0), (1,-1), (1,-2), (0,0), (0,-1), (0,-2), (-1,-1), and (-1,-2). - _Danny Rorabaugh_, Apr 05 2016

David A. Corneth, Table of n, a(n) for n = 0..9999

Cf. A000005, A067274, A226355.

a[n_]:=2Sum[Length[Divisors[k]],{k,1,n}]+Floor[Sqrt[n]]+2n+1

a(n) = 2*sum(k=1, n, numdiv(k)) + sqrtint(n) + 2*n + 1; \\ Michel Marcus, Apr 05 2016

a(n) = 2*(Sum_{k=1..n} tau(k)) + floor(sqrt(n)) + 2*n + 1, where tau(k) = A000005(k) is number of divisors of k.

a(n) = A067274(n) + 2 for n >= 1.

A269067 Numerator of the volume of d dimensional symmetric convex cuboid with constraints |x1 + x2 + ... xd| <= 1 and |x1|, |x2|, ..., |xd| <= 1.

Original entry on oeis.org

2, 3, 16, 115, 88, 5887, 19328, 259723, 124952, 381773117, 41931328, 20646903199, 866732192, 467168310097, 2386873693184, 75920439315929441, 97261697912, 5278968781483042969, 2387693641959232, 9093099984535515162569, 10872995484706511008, 168702835448329388944396777, 38650653745373963289088

Offset: 1

Lorenz H. Menke, Jr., Feb 19 2016

Reference A. Dubickas shows that all the volume integrals are rational with V[d] <= 2^d.

			For d = 3 the volume is 16/3, for each volume we have V[1] = 2, V[2] = 3, V[3] = 16/3, V[4] = 115/12, V[5] = 88/5, V[6] = 5887/180, V[7] = 19328/315, V[8] = 259723/2240, V[9] = 124952/567, V[10] = 381773117/907200, etc.

R. Chela, Reducible Polynomials, Journal London Math. Soc. 38 (1963), pp 183-188 Eq. 7.
Arturas Dubickas, On the number of reducible polynomials of bounded naive height, Manuscripta Math. 144 (2014), pp 439-456, Eq. 4, 5 & Section 5.
Mathematica Stack Exchange, How to improve or optimize a volume integration over a cuboid

The denominator sequence is given by A266913.

Cf. A199832 The rational coefficient of the leading coefficient of the empirical rows duplicates these volume integrals in sequence. This is not a proof.

V[d_] := Integrate[Boole[Abs[Sum[x[i], {i, 1, d}]] <= 1],
  Table[x[i], {i, 1, d}] \[Element]
   Cuboid[Table[-1, {i, 1, d}], Table[+1, {i, 1, d}]]] (* Lorenz H. Menke, Jr. *)
v[d_] := With[{a = Array[x,d]}, RegionMeasure @ ImplicitRegion[ a ∈ Cuboid[-Table[1, d], Table[1, d]] && -1 <= Total[a] <= 1, a]] (* Carl Woll *)
v[d_] := 2^(d+1)/(Pi) Integrate[Sin[t]^(d+1)/t^(n+1), {t, 0, Infinity}] (* Carl Woll *)

a(11)-a(23) from Lorenz H. Menke, Jr., May 10 2018

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User: Lorenz H. Menke, Jr.

A358399 a(n) is the number of reducible monic quartic polynomials (x^4 + r*x^3 + s*x^2 + t*x + u) with integer coefficients bounded by naïve height n (abs(r), abs(s), abs(t), abs(u) <= n).

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A358400 a(n) is the number of reducible monic quintic polynomials (x^5 + r*x^4 + s*x^3 + t*x^2 + u*x + v) with integer coefficients bounded by naïve height n (abs(r), abs(s), abs(t), abs(u), abs(v) <= n).

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A358398 a(n) is the number of reducible monic cubic polynomials x^3 + r*x^2 + s*x + t with integer coefficients bounded by naïve height n (abs(r), abs(s), abs(t) <= n).

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A281914 Number of ordered integer pairs (b,c), with -n<=b<=n, -n<=c<=n, such that both roots of 3x^2+bx+c = 0 are rational and b and are not both multiples of 3.

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A281913 Number of ordered integer pairs (b,c), with -n<=b<=n, -n<=c<=n, such that both roots of 2x^2+bx+c=0 are rational and b and c are not both even.

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A281376 Total number of counts where floor(N/k) < floor((N+k)/n) for k = {1, 2, ..., n-1} and N >= n.

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A270544 Number of ordered pairs (i,j) with |i|, |j| <= n, |i * j| <= n, and i odd.

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A270543 Number of ordered pairs (i,j) of integers with |i|, |j| <= n, |i * j| <= n, and i even.

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Extensions

A270383 Number of ordered pairs (i,j) with i >= j, |i|, |j| <= n, and |i * j| <= n.

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A358399 a(n) is the number of reducible monic quartic polynomials (x^4 + rx^3 + sx^2 + t*x + u) with integer coefficients bounded by naïve height n (abs(r), abs(s), abs(t), abs(u) <= n).

A358400 a(n) is the number of reducible monic quintic polynomials (x^5 + rx^4 + sx^3 + tx^2 + ux + v) with integer coefficients bounded by naïve height n (abs(r), abs(s), abs(t), abs(u), abs(v) <= n).

A358398 a(n) is the number of reducible monic cubic polynomials x^3 + rx^2 + sx + t with integer coefficients bounded by naïve height n (abs(r), abs(s), abs(t) <= n).