Giedrius Alkauskas - cp's OEIS frontend

A386302 Positive integers k such that the set {d+k/d : d|k} contains three consecutive integers.

144, 180, 1260, 1440, 2520, 5040, 5544, 7200, 14040, 15120, 25200, 31680, 33660, 37800, 46800, 59400, 62244, 65520, 70560, 83160, 107100, 110880, 115920, 166320, 169344, 176400, 180180, 183600, 190944, 221760, 277200, 287280, 297540

Offset: 1

Giedrius Alkauskas, Jul 17 2025

nonn

Terms are divisible by 36.

Subsequence of A072389 (with two consecutive rather than three).

			a(1)=144, since 144/12+12=24, 144/9+9=25, 144/8+8=26, and no smaller integer with such property exists.

Giedrius Alkauskas, Consecutive integers in the set S_n={d+n/d: d|n}

Cf. A002496, A027862, A072389, A335572, A386303.

M:=300000:
Ki:={}:
Vi:=floor(sqrt(2*M)):
Ski:=floor((19*M)^(1/4)/2):
for F from 1 to Vi-4 do
  for y from 1 to min(floor((Vi-F)/2),Ski) do
     G:=F+2*y+1:
     if issqr(2*F^2-G^2+2) then
       x:=(F+G-1)/2;
       n:=x*(x+1)*y*(y+1):
       Ki:=Ki union {n}:
     end if:
  end do:
end do:
Ki;

isok(m, nb=3) = nb--; my(v = Set(apply(x->x+m/x, divisors(m)))); if (#v >= nb, select(x->(x==nb), vector(#v-nb, k, v[k+nb]-v[k]))); \\ Michel Marcus, Jul 18 2025

A386303 Positive integers k such that the set {d+k/d : d|k} contains four consecutive integers.

Original entry on oeis.org

15120, 712800, 3341520, 10533600, 23284800, 85503600, 147026880, 171097920, 302702400, 477338400, 2058376320, 2633510880, 4204418400, 7342876800, 9673606800, 13035884400, 13734761040, 14895223200, 22388788800, 22647794400, 26108082000, 34183749600, 62246804400, 89169141600

Offset: 1

Giedrius Alkauskas, Jul 18 2025

nonn

a(n) is divisible by 720.
Subsequence of A072389 (with two consecutive instead of four).
Integers k with five consecutive integers in the set {d+k/d : d|k} seem not to exist.
As terms must be of the form k * (k + 1) * m * (m + 1) and divisible by 720 we can restrict the search based on g = gcd(k * (k + 1), 720) which is at least 2. We must have (720 / g) | m * (m + 1). - David A. Corneth, Jul 19 2025
If q is the number of divisors of a(n) then the first of these four divisors is generally d[q/2 + 1] at least for nonsquares. For three consecutive integers (cf. A386302) there is the exception 180180. - David A. Corneth, Jul 20 2025

			a(1)=15120=M is a term of this sequence since 105, 108, 112, 120 are divisors of M, and 120+M/120=246, 112+M/112=247, 108+M/108=248, 105+M/105=249. It is the first term since no smaller such positive integer exists.

Giedrius Alkauskas, Consecutive integers in the set S_n={d+n/d: d|n}
David A. Corneth, 360 x 360 pixels image where white pixels with coordinate (k, m) have 720 | (k * (k + 1) * m * (m + 1))
David A. Corneth, PARI program

Cf. A002496, A027862, A072389, A335572, A386302.

M:=2*10^10:
Ki:={}:
Vi:=floor(sqrt(2*M)):
Ski:=floor((19*M)^(1/4)/2):
for F from 1 to Vi-4 do
  for y from 1 to min(floor((Vi-F)/2),Ski) do
     G:=F+2*y+1:
     if issqr(2*F^2-G^2+2) and issqr(3*F^2-2*G^2+6) then
       x:=(F+G-1)/2:
       n:=x*(x+1)*y*(y+1):
       Ki:=Ki union {n}:
     end if:
  end do:
end do:
Ki;

More terms from David A. Corneth, Jul 19 2025

A375986 Maximum number of edges in a simple polygon that is the union of n triangles.

Original entry on oeis.org

3, 12, 22

Offset: 1

Giedrius Alkauskas, Sep 07 2024

By an explicit construction, a(n) >= 11*n-11 for all n >= 4.

			For n = 1, one has three vertices of a single triangle.
For n = 2 the example is a hexagram (the star of David).

Giedrius Alkauskas, Construction which shows that for n >= 4, a(n) >= 11*n-11.
Math Stackexchange, If a union of n triangles is a simple M-gon, find the maximum of M.

A369382 Number of subsets of the integer lattice Z^2 of cardinality n such that there is no monotone lattice path which splits the set in half, up to lattice symmetry.

Original entry on oeis.org

0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 2, 0, 2, 0, 2, 0, 4, 0, 4, 1, 3, 0, 2, 0, 3, 0, 6, 0, 10, 0, 6, 0, 9, 0, 12, 1, 18, 2, 9, 0, 5, 0, 7, 0, 8, 0, 12, 0, 18, 0, 14, 0, 17

Offset: 1

Giedrius Alkauskas, Jan 22 2024

A monotone path is a lattice path consisting of east and north unit steps or a path consisting of east and south unit steps. When counting, points lying on the path itself are discarded.
Related to A367783, only sets obtained by rotation and reflection are considered to be the same.
For odd n, a(n) = A367783(n)/8.
For even n, 8 * a(n) >= A367783(n).
a(n) > 0 for even n >= 12.
a(n) > 0 for odd n with natural density 1 (among odd numbers).

			For n = 4, a(4) = 1 way to place 4 points is as follows:
.xx.
.xx.
For n = 14, a(14) = 1 way to place 14 points is as follows:
  ...x..
  ..x.x.
  .xxx.x
  x.xxx.
  .x.x..
  ..x...
For n = 27, a(27) = 1 way to place 27 points is as follows:
  ....x....
  ...x.....
  ..x......
  .x..xx...
  x..xxxx..
  ..xxxxxxx
  ...xxxxx.
  ....xxx..
  .....x...

Giedrius Alkauskas, Friendly paths for finite subsets of plane integer lattice. I, arXiv:2302.01137 [math.CO], 2024.
Giedrius Alkauskas, Problem 11484, Problems and solutions, Amer. Math. Monthly, 117 (2) February (2010), p. 182.
Giedrius Alkauskas, Friendly paths. Problem 11484, Problems and solutions, Amer. Math. Monthly, 119 (2) February (2012), 167-168.

Cf. A000009, A000292, A005232, A367783.

A367783 Number of subsets of the integer lattice Z^2 of cardinality n such that there is no monotone lattice path which splits the set in half, up to shifts.

Original entry on oeis.org

0, 0, 0, 1, 0, 0, 0, 2, 0, 0, 0, 1, 0, 2, 0, 2, 0, 4, 0, 6, 0, 6, 0, 11, 0, 12, 8, 7, 0, 6, 0, 8, 0, 18, 0, 32, 0, 20, 0, 29, 0, 42, 8, 67, 16, 30, 0, 13, 0, 22, 0, 32, 0, 42, 0, 64, 0, 50, 0, 64

Offset: 1

Giedrius Alkauskas, Nov 30 2023

A monotone path is a lattice path consisting of east and north unit steps or a path consisting of east and south unit steps. When counting, points lying on the path itself are discarded.
a(n) > 0 for even n >= 12.
a(n) > 0 for odd n with natural density 1 (among odd numbers).
For odd n, a(n) is divisible by 8.

			For n = 4 a(4) = 1 way to place 4 points is as follows:
  .xx.
  .xx.
For n = 8 a(8) = 2 ways to place 8 points are as follows:
  ..x.
  .xxx
  xxx.
  .x..
(and its reflection with respect to a vertical axis).
For n = 18 a(18) = 4 ways to place 18 points are as follows:
  ...x..
  ..xxx.
  .xxxxx
  xxxxx.
  .xxx..
  ..x...
(and its reflection with respect to a vertical axis), and
  .....x....
  ......x...
  .......x..
  ....x...x.
  ...xxx...x
  x...xxx...
  .x...x....
  ..x.......
  ...x......
  ....x.....
(and its reflection with respect to a vertical axis).

Giedrius Alkauskas, Friendly paths for finite subsets of plane integer lattice. I, arXiv:2302.01137 [math.CO], 2024.
Giedrius Alkauskas, Problem 11484, Problems and solutions, Amer. Math. Monthly, 117 (2) February (2010), p. 182.
Giedrius Alkauskas, Friendly paths. Problem 11484, Problems and solutions, Amer. Math. Monthly, 119 (2) February (2012), 167-168.

Cf. A000009, A000292, A005232, A369382.

a(36) corrected by Giedrius Alkauskas, Feb 02 2024

a(49)-a(60) from Giedrius Alkauskas, Feb 06 2024

A360427 Values of the argument at successive record minima of the function R defined as follows. For any integer x >= 1, let y > x be the smallest integer such that there exist integers x < c < d < y such that x^3 + y^3 = c^3 + d^3. Then R(x) = y/x.

Original entry on oeis.org

1, 2, 8, 9, 10, 17, 30, 42, 51, 135, 156, 285, 792, 1634, 3751, 4026, 6192, 14934, 15768, 16147, 45121, 58230, 61389, 79876, 167757, 177560, 213652, 525537, 917324, 1050787, 2237052, 3954983, 4157802

Offset: 1

Giedrius Alkauskas, Feb 07 2023

For a given integer x, the identity x^3 + (12x)^3 = (9x)^3 + (10x)^3 holds, so R(x) <= 12.

A quadruple x = 2*N^4 - 4*N^3 + 9*N^2 - 8*N +10, y = 2*N^4 + 6*N^2 + N + 9, c = 2*N^4 - 3*N^3 + 12*N^2 - 5*N + 12, d = 2*N^4 - N^3 + 6*N^2 + N + 1 (for integer N) shows that the sequence is infinite.

			For x = 1, y = 12, 1^3 + 12^3 = 9^3 + 10^3, R(1) = 12. So, a(1) = 1.
For x = 2, y = 16, 2^3 + 16^3 = 9^3 + 15^3, R(2) = 8. So, a(2) = 2.
For x = 3, y = 36, 3^3 + 36^3 = 27^3 + 30^3, R(3) = 12. So, this does not provide a record minimum. The same negative outcome happens for x = 4, x = 5, x = 6, x = 7.
For x = 8, y = 53, 8^3 + 53^3 = 29^3 + 50^3, R(8) = 6.625. So, a(4) = 8.
For n = 8, a(8) = 42, since 42^3 + 69^3 = 56^3 + 61^3, and the ratio R(42) = 69/42 = 1.6428571... is an absolute minimum (eighth successive) for the function R(x) for 1 <= x <= 42.

Giedrius Alkauskas, On the sequence representing narrowest intervals containing two pairs of integers with equal cube sums.
Ajai Choudhry, On equal sums of cubes, Rocky Mountains Journal of Mathematics, 28 (4), 1998.

Cf. A001235, A011541.

xm,ym,x,n = 0,1,0,1
while True:
    x,y = x+1,x+4
    while y*xm < ym*x:
        c,d,s = x+1,y-1,x**3+y**3
        while cs:
                d-=1
            else:
                break
        if t==s:
            print("a({})={} x={} c={} d={} y={}".format(n,x,x,c,d,y))
            xm,ym,n = x,y,n+1
            break
        y+=1
# Bert Dobbelaere, Mar 18 2023

a(25)-a(33) from Bert Dobbelaere, Mar 18 2023

A360619 a(n) > n is the smallest integer such that there exist integers n < c < d satisfying n^3 + a(n)^3 = c^3 + d^3.

Original entry on oeis.org

12, 16, 36, 32, 60, 48, 84, 53, 34, 27, 93, 40, 156, 112, 80, 106, 39, 68, 228, 54, 238, 176, 94, 80, 167, 156, 102, 224, 99, 67, 246, 166, 279, 78, 98, 120, 174, 304, 468, 108, 319, 69, 516, 352, 170, 188, 97, 160, 282, 96, 82, 312, 550, 204, 113, 371, 180, 198, 708, 134, 600

Offset: 1

Giedrius Alkauskas, Feb 14 2023

nonn

Since the identity n^3 + (12n)^3 = (9n)^3 + (10n)^3 holds, n < a(n) <= 12n.

			For n = 11, a(11) = 93, since, first, 11^3 + 93^3 = 30^3 + 92^3. Second, for any integral y in the range [12, 92] there does not exist c, d, 11 < c < d < y, satisfying 11^3 + y^3 = c^3 + d^3.

Jean-François Alcover, Table of n, a(n) for n = 1..200 (terms 1..61 from Giedrius Alkauskas).
Giedrius Alkauskas, On the sequence representing narrowest intervals containing two pairs of integers with equal cube sums.

Cf. A360427, A001235.

a :=proc(n::integer) local found::boolean; local N, SQ, i;
found:=false; N:=n+1; SQ:={};
while not found do SQ:=SQ union {N^3}; N:=N+1;
for i from n+1 to N-1 do if evalb(N^3+n^3-i^3 in SQ) then
found:=true; end if; end do; end do; N end proc;

a[n_] := a[n] = Module[{found, m, SQ, i}, found = False; m = n+1; SQ = {}; While[!found, SQ = SQ ~Union~ {m^3}; m = m+1; For[i = n+1, i <= m-1, i++, If[MemberQ[SQ, m^3+n^3-i^3], found = True]]]; m];
Table[Print[n, " ", a[n]]; a[n], {n, 1, 200}] (* Jean-François Alcover, Feb 27 2023, after Giedrius Alkauskas's Maple code *)

A360796 a(n) > n is the smallest integer such that there exist integers n < c <= d < a(n) satisfying n^2 + a(n)^2 = c^2 + d^2.

Original entry on oeis.org

7, 9, 11, 13, 14, 17, 17, 19, 20, 25, 23, 29, 26, 27, 29, 37, 31, 40, 34, 35, 38, 46, 39, 41, 44, 43, 44, 54, 47, 58, 49, 51, 56, 53, 54, 67, 62, 59, 59, 70, 62, 73, 64, 65, 74, 78, 69, 71, 71, 75, 74, 86, 76, 77, 79, 83, 92, 93, 83, 103

Offset: 1

Giedrius Alkauskas, Feb 21 2023

nonn

n^2 + a(n)^2 belongs to A007692.
The identity n^2 + (2*n + 5)^2 = (n+4)^2 + (2*n + 3)^2 shows that a(n) <= 2*n + 5. The last case when the equality holds is n = 16.
a(n) = a(n+1) has infinitely many solutions. This holds, in particular, when n = (u*v + u + v - 1) * (u*v - 2)/2 - 1 for positive integers u, v satisfying v+2 <= u <= 6*v - 3.
a(n-1) = a(n) = a(n+1) holds for n = (3*v^2 + 5*v + 1) * (6*v^2 + 3*v - 2), v >= 3.

			a(10) = 25, since 10^2 + 25^2 = 14^2 + 23^2, and no integers b, c, d exist satisfying 10 < c <= d < b < 25 and 10^2 + b^2 = c^2 + d^2.

Giedrius Alkauskas, On the function which is defined by the minimal solution to n^2 + f(n)^2 = a^2 + b^2, a, b, f(n) > n.

Cf. A360619, A000404, A007692.

a :=proc(n::integer) local found::boolean; local N, SQ, i;
found:=false; N:=n+1; SQ:={};
while not found do SQ:=SQ union {N^2}; N:=N+1;
for i from n+1 to N-1 do
if evalb(N^2+n^2-i^2 in SQ) then found:=true; end if;
end do; end do; N end proc;

A358212 a(n) is the maximal possible sum of squares of the side lengths of an n^2-gon supported on a subset 1 <= x,y <= n of an integer lattice.

Original entry on oeis.org

4, 10, 36, 98, 232

Offset: 2

Giedrius Alkauskas, Nov 04 2022

Examples show that a(7) >= 462, a(8) >= 842, a(9) >= 1424, a(10) >= 2242.

Asymptotics: liminf a(n)/n^4 >= 8/27, limsup a(n)/n^4 <= 2/3.

Oliver Mantas Ališauskas, Grid connector, Web application for this problem.
Oliver Mantas Ališauskas, Giedrius Alkauskas, and Valdas Dičiūnas, Full Grid Lattice Polygons with Maximal Sum of Squares of Edge-Lengths, arXiv:2311.03011 [math.CO], 2023-2024.
S. Chow, A. Gafni, and P. Gafni, Connecting the dots: maximal polygons on a square grid, Math. Mag. 94 (2021), no. 2, 118-124.
G. L. Cohen and E. Tonkes, Dartboard arrangements, Elect. J. Combin., 8(2) (2001), #R4.

Cf. A209077, A110611, A064842, A226595, A226596.

a(5) from Giedrius Alkauskas, Oct 09 2023

a(6) from Giedrius Alkauskas, Nov 30 2023

A354673 Smallest number of unit cells that must be removed from an n X n square board in order to avoid any cycles.

Original entry on oeis.org

0, 1, 2, 4, 6, 10, 13, 18, 22, 28, 34, 42, 49, 58, 66, 76, 86, 98, 109, 122, 134, 148, 162, 178, 193, 210, 226, 244, 262, 282, 301, 322, 342, 364, 386, 410, 433, 458, 482, 508, 534, 562, 589, 618, 646, 676, 706, 738, 769, 802, 834, 868, 902, 938, 973, 1010, 1046

Offset: 1

Giedrius Alkauskas, Jun 02 2022

A "cycle" means a rook-connected closed path of squares.
The proof of this result is given in the Links section.
a(n+1) is very close to A239231(n); more precisely, the difference is the sequence 1,0,1,1,1,1,1,1,1,1,1,2,1,1,1,2,1,1,1,2,1,2,3,2.

			For n = 2, a(2) = 1, since removing any unit square from the 2 X 2 board leaves no cycles.
For n = 5, a(5) = 6 removed unit squares can be arranged as follows:
  x****
  *x*x*
  **x**
  *x*x*
  *****

Giedrius Alkauskas, Maximal subsets of n X n board without cycles, 2022.
Index entries for linear recurrences with constant coefficients, signature (2,-1,0,0,0,1,-2,1).

Cf. A085577, A239072, A085577, A104519, A000170, A239231.

a(n) = ceiling(n^2/3 - n/6 + 4/3) - ceiling(n/2) for n >= 3.
From Stefano Spezia, Jun 02 2022: (Start)
G.f.: x^2*(1 + x^2 + 2*x^4 - x^5 + x^6 - x^7 + x^8)/((1 - x)^3*(1 + x)*(1 - x + x^2)*(1 + x + x^2)).
a(n) = 2*a(n-1) - a(n-2) + a(n-6) - 2*a(n-7) + a(n-8) for n > 2. (End)

cp's OEIS Frontend

User: Giedrius Alkauskas

A386302 Positive integers k such that the set {d+k/d : d|k} contains three consecutive integers.

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A386303 Positive integers k such that the set {d+k/d : d|k} contains four consecutive integers.

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A375986 Maximum number of edges in a simple polygon that is the union of n triangles.

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A369382 Number of subsets of the integer lattice Z^2 of cardinality n such that there is no monotone lattice path which splits the set in half, up to lattice symmetry.

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A367783 Number of subsets of the integer lattice Z^2 of cardinality n such that there is no monotone lattice path which splits the set in half, up to shifts.

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Extensions

A360427 Values of the argument at successive record minima of the function R defined as follows. For any integer x >= 1, let y > x be the smallest integer such that there exist integers x < c < d < y such that x^3 + y^3 = c^3 + d^3. Then R(x) = y/x.

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Extensions

A360619 a(n) > n is the smallest integer such that there exist integers n < c < d satisfying n^3 + a(n)^3 = c^3 + d^3.

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A360796 a(n) > n is the smallest integer such that there exist integers n < c <= d < a(n) satisfying n^2 + a(n)^2 = c^2 + d^2.

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