W. Edwin Clark - cp's OEIS frontend

A376691 The number of solutions to x + y == x^2 + y^2 == x^3 + y^3 (mod n) with 0 <= x <= y < n.

1, 3, 3, 5, 3, 10, 3, 7, 5, 10, 3, 16, 3, 10, 10, 7, 3, 18, 3, 16, 10, 10, 3, 24, 7, 10, 5, 16, 3, 36, 3, 11, 10, 10, 10, 28, 3, 10, 10, 24, 3, 36, 3, 16, 18, 10, 3, 24, 9, 26, 10, 16, 3, 18, 10, 24, 10, 10, 3, 56, 3, 10, 18, 11, 10, 36, 3, 16, 10, 36

Offset: 1

W. Edwin Clark, Oct 01 2024

nonn

This is the same as the number of solutions to x + y == x^2 + y^2 == x^3+y^3 == x^4 + y^4 (mod n) with x <=y. Proved by Sahaj in Math StackExchange link.

Math StackExchange, If x+y == x^2+y^2 == x^3+y^3 (mod n), is it true that x+y==x^4+y^4(mod n)?.

Cf. A376646.

a:=proc(n)
 local x,y,count;
  count:=0:
  for x from 0 to n-1 do
   for y from x to n-1 do
     if (x+y) mod n =(x^2+y^2) mod n and (x+y) mod n =(x^3+y^3) mod n then count:=count+1;  fi;
   od:
  od:
  count;
end:

def A376691(n):
    c = 0
    for x in range(n):
        m = x*(1-x)%n
        c += sum(1 for y in range(x,n) if y*(y-1)%n == m and not m*(x-y)%n)
    return c # Chai Wah Wu, Oct 02 2024

A376646 Number of solutions to x + y == x^2 + y^2 (mod n) with x <= y.

Original entry on oeis.org

1, 3, 3, 6, 3, 10, 5, 10, 7, 10, 7, 20, 7, 18, 10, 18, 9, 26, 11, 20, 18, 26, 13, 36, 11, 26, 19, 36, 15, 36, 17, 34, 26, 34, 18, 52, 19, 42, 26, 36, 21, 68, 23, 52, 26, 50, 25, 68, 29, 42, 34, 52, 27, 74, 26, 68, 42, 58, 31, 72, 31, 66, 50, 66, 26, 100, 35, 68, 50, 68

Offset: 1

W. Edwin Clark, Sep 30 2024

Alois P. Heinz, Table of n, a(n) for n = 1..4000

a:=proc(n)
 local x,y,count;
  count:=0:
  for x from 0 to n-1 do
   for y from x to n-1 do
     if (x+y) mod n =(x^2+y^2) mod n  then count:=count+1; fi;
   od:
  od:
  count;
end:
# second Maple program:
a:= n-> add(add(`if`(x^2-x+y^2-y mod n=0, 1, 0), x=0..y), y=0..n-1):
seq(a(n), n=1..70);  # Alois P. Heinz, Oct 01 2024

a(n) = sum(y=0, n-1, sum(x=0, y, (x+y) % n == (x^2+y^2) % n)); \\ Michel Marcus, Oct 01 2024

def A376646(n):
    c = 0
    for x in range(n):
        m = x*(1-x)%n
        c += sum(1 for y in range(x,n) if y*(y-1)%n==m)
    return c # Chai Wah Wu, Oct 02 2024

A376190 For a line L in the plane, let C(L) denote the number of prime points [k, prime(k)] on L, and let M(L) denote the maximum prime(k) for any of these points. a(n) is the maximum of the smallest primes in the lines L with C(L) = n and containing prime A376187(n), or a(n) = -1 if no such lines exist.

Original entry on oeis.org

2, 2, 3, 5, 19, 18, 7, 13, 967, 113, 83, 619, 103, 1583, 1693, 1621, 1759, 1753, 5923, 197, 6143, 15823, 5849, 1609, 1663, 10333, 1613, 152003, 15683, 16111, 1619, 141871, 15649, 15383, 140989, 141811, 136481, 141319, 15667, 136769, 16033, 16619, 141707, 15473, 135649

Offset: 1

N. J. A. Sloane, Sep 25 2024, following a suggestion from W. Edwin Clark

Consider all the lines L in the plane containing exactly n prime-points (k, prime(k)). A376187 minimizes the maximal prime on any such line L, while the present sequence then maximizes the minimal prime on the lines from A376187.
In other words, we first minimize (in A376187) the maximal prime over all lines with exactly n points, and then here we further maximize the minimal prime. The second step minimizes the spread of the points.
For most listed terms, there is only one line L with C(L) = n that contains prime A376187(n). - Max Alekseyev, Sep 28 2024

			The best line with 5 points contains the primes 19,23,31,43,47, so a(5) = 19 and A376187(5) = 47. See the Table for further examples.

N. J. A. Sloane, Table of lines in the plane containing the known maximum numbers of prime-points

Cf. A376187, A376188.

Better definition and a(28)-a(45) from Max Alekseyev, Sep 28 2024

A376318 The number of distinct values of x+y+z (mod n) when xyz = 1 (mod n).

Original entry on oeis.org

1, 1, 2, 1, 4, 2, 7, 2, 4, 4, 11, 2, 13, 7, 8, 3, 17, 4, 19, 4, 14, 11, 23, 4, 20, 13, 11, 7, 29, 8, 31, 6, 22, 17, 28, 4, 37, 19, 26, 8, 41, 14, 43, 11, 16, 23, 47, 6, 49, 20, 34, 13, 53, 11, 44, 14, 38, 29, 59, 8, 61, 31, 28, 11, 52, 22, 67, 17, 46, 28, 71, 8, 73, 37, 40, 19, 77, 26

Offset: 1

W. Edwin Clark, Sep 22 2024

nonn

The values of n for which a(n) = n seem to be A007775, but I have no proof of this.

Charles R Greathouse IV, Table of n, a(n) for n = 1..30000
Charles R Greathouse IV, Program for computing terms (C with PARI)

Cf. A007775, A376296, A376183, A180783, A376427.

a:=proc(n)
local x,y,z,N;
N:=NULL;
for x from 0 to n-1 do
 for y from x to n-1 do
  for z from y to n-1 do
   if (x*y*z) mod n = 1 mod n then N:=N,(x+y+z) mod n; fi;
  od:
 od:
od:
nops({N});
end:

a(n)=my(v=vectorsmall(n)); for(x=1,n, if(gcd(x,n)>1, next); for(y=1,x, if(gcd(y,n)>1, next); my(z=1/Mod(x*y,n)); v[lift(x+y+z)+1]=1)); sum(i=1,n, v[i]) \\ Charles R Greathouse IV, Sep 23 2024

def A376318(n):
    s = set()
    for x in range(n):
        for y in range(x,n):
            try:
                s.add((x+y+pow(x*y%n,-1,n))%n)
            except:
                continue
    return len(s) # Chai Wah Wu, Sep 23 2024

A376427 The number of distinct values of x+y+z+w (mod n) when xyz*w = 1 (mod n).

Original entry on oeis.org

1, 1, 3, 1, 5, 3, 7, 2, 5, 5, 11, 3, 13, 7, 15, 4, 17, 5, 19, 5, 21, 11, 23, 6, 25, 13, 15, 7, 29, 15, 31, 8, 33, 17, 35, 5, 37, 19, 39, 10, 41, 21, 43, 11, 25, 23, 47, 12, 49, 25, 51, 13, 53, 15, 55, 14, 57, 29, 59, 15, 61, 31, 35, 16, 65, 33, 67, 17, 69, 35, 71, 10, 73, 37, 75, 19, 77, 39, 79, 20, 45, 41, 83, 21, 85, 43, 87, 22, 89, 25

Offset: 1

W. Edwin Clark, Sep 22 2024

nonn

The values of n for which a(n) = n seem to agree with A325128. But I have no proof.

Chai Wah Wu, Table of n, a(n) for n = 1..1688

Cf. A376296, A376183, A180783.

a:=proc(n)
local x,y,z,w,N;
N:={};
for x from 0 to n-1 do
 for y from x to n-1 do
  for z from y to n-1 do
   for w from z to n-1 do
     if (x*y*z*w) mod n = 1 mod n then N:=N union {(x+y+z+w) mod n}; fi;
   od:
  od:
 od:
od:
nops(N);
end:

def A376427(n):
    s = set()
    for x in range(n):
        for y in range(x,n):
            xy, xyp = x*y%n, (x+y)%n
            for z in range(y,n):
                try:
                    s.add((xyp+z+pow(xy*z%n,-1,n))%n)
                except:
                    continue
    return len(s) # Chai Wah Wu, Sep 23 2024

A376296 The number of solutions x<=y<=z<=w in Z/(n) of the equation x+y+z+w = xyz*w.

Original entry on oeis.org

1, 2, 6, 7, 14, 18, 27, 34, 51, 59, 91, 96, 134, 136, 208, 203, 285, 261, 385, 373, 493, 487, 650, 616, 818, 750, 949, 947, 1240, 1146, 1517, 1397, 1766, 1662, 2089, 1824, 2443, 2309, 2723, 2638, 3311, 2977, 3801, 3482, 4024, 3962, 4900, 4382, 5525, 5023, 6078

Offset: 1

W. Edwin Clark, Sep 19 2024

nonn

Chai Wah Wu, Table of n, a(n) for n = 1..1351

Cf. A376183, A180783.

a:=proc(n)
local x,y,z,w,N;
N:=0:
for x from 0 to n-1 do
 for y from x to n-1 do
  for z from y to n-1 do
   for w from z to n-1 do
    if (x+y+z+w-x*y*z*w) mod n = 0 then N:=N + 1; fi;
   od:
  od:
 od:
od:
N;
end:

def A376296(n):
    c = 0
    for x in range(n):
        for y in range(x,n):
            xy,xyp = x*y%n,(x+y)%n
            for z in range(y,n):
                xyz, xyzp = xy*z%n-1,(xyp+z)%n
                c += sum(not (xyz*w-xyzp)%n for w in range(z,n))
    return c # Chai Wah Wu, Sep 19 2024

A376183 The number of solutions x<=y<=z in Z/(n) of the equation x+y+z = xyz.

Original entry on oeis.org

1, 3, 2, 4, 7, 8, 10, 13, 13, 31, 24, 20, 37, 44, 38, 47, 59, 59, 66, 86, 53, 108, 96, 77, 137, 171, 100, 120, 159, 186, 170, 179, 135, 279, 230, 172, 253, 312, 220, 337, 307, 259, 322, 306, 331, 456, 384, 303, 369, 669, 366, 500, 503, 488, 588, 469, 409, 767, 600

Offset: 1

W. Edwin Clark, Sep 14 2024

nonn

Suggested by a discussion initiated by Keith F. Lynch on the MathFun mailing list Sept 8, 2024 about when sums and products of real numbers x,y,z are integers and later raising other similar questions.

Chai Wah Wu, Table of n, a(n) for n = 1..8450 (terms 1..2000 from Hugo Pfoertner)

a:=proc(n)
local x,y,z,w,N;
N:=0:
for x from 0 to n-1 do
 for y from x to n-1 do
  for z from y to n-1 do
   if (x+y+z-x*y*z) mod n = 0 then N:=N + 1; fi;
  od:
 od:
od:
N;
end:

a(n) = sum(x=0, n-1, sum(y=x, n-1, sum(z=y, n-1, Mod(x+y+z-x*y*z, n)==0))); \\ Michel Marcus, Sep 15 2024

def A376183(n):
    c = 0
    for x in range(n):
        for y in range(x,n):
            xy,xyp = x*y%n-1,(x+y)%n
            c += sum(not (xy*z-xyp)%n for z in range(y,n))
    return c # Chai Wah Wu, Sep 19 2024

More terms from Hugo Pfoertner, Sep 15 2024

A375625 Number of distinct values taken by F(p) = Sum (|i-j| - |p(i)-p(j)|)^2 where the sum is over all 1 <= i < j <= n, for all permutations p in the symmetric group S_n.

Original entry on oeis.org

1, 1, 1, 2, 4, 10, 17, 51, 55, 160, 140, 389, 300, 795, 566, 1290

Offset: 0

W. Edwin Clark, Aug 21 2024

The function F was defined by Dan Asimov on the Mailing list Math-Fun on Aug. 18, 2024.

Cf. A126972, A375623.

F:= S-> add(add((j-i-abs(S[j]-S[i]))^2, i=1..j-1), j=2..nops(S)):
a:= n-> nops({map(F, combinat[permute](n))[]}):
seq(a(n), n=0..10);

a(11)-a(13) from Hugo Pfoertner, Aug 24 2024

a(14)-a(15) from Hugo Pfoertner, Sep 04 2024

A375623 Maximum value of F(p) = Sum (|i-j| - |p(i)-p(j)|)^2 where the sum is over all 1 <= i < j <= n, for all permutations p in the symmetric group S_n.

Original entry on oeis.org

0, 0, 0, 2, 12, 30, 72, 132, 240, 380, 600, 870, 1260, 1722, 2532, 3080

Offset: 0

W. Edwin Clark, Aug 21 2024

The function F was defined by Dan Asimov on the Mailing list Math-Fun on Aug. 18, 2024. It can be considered as a sort of entropy of a permutation p like the function Sum_{k=1..n} (p(k)-k)^2 in A126972.

The terms for even n seem to agree with A047928.

Cf. A047928, A126972, A375625.

F := proc(S) local i, j, M;
M := 0;
for j from 1 to nops(S) do
    for i from 1 to j-1 do
      M := M + (abs(i - j) - abs(S[i] - S[j]))^2
    od:
od: M end:
a := proc(n) local P, m, u, mm;
P := combinat:-permute(n);
m := 0;
for u in P do
   mm := F(u);
   if mm > m then m := mm fi;
od: m end:

a375623(n) = my(m=0); forperm(n, p, m=max(m, sum(i=1,n, sum(j=1,i-1,(abs(i-j)-abs(p[i]-p[j]))^2)))); m \\ Hugo Pfoertner, Aug 22 2024

a(11)-a(13) from Hugo Pfoertner, Aug 23 2024
a(14) from Markus Sigg, Aug 25 2024
a(15) from Hugo Pfoertner, Sep 04 2024

A346663 The number of nonreal roots of Sum_{k=0..n} prime(k+1)*x^k.

Original entry on oeis.org

0, 0, 2, 2, 4, 4, 6, 6, 8, 8, 10, 10, 12, 12, 14, 14, 16, 16, 18, 18, 20, 20, 22, 22, 24, 24, 26, 26, 28, 28, 30, 30, 32, 32, 34, 34, 36, 36, 38, 38, 40, 40, 42, 42, 44, 44, 46, 46, 48, 48, 50, 50, 52, 52, 54, 54, 56, 56, 58, 58, 60, 60, 62, 62, 64, 64, 66, 66, 68, 68, 70, 70, 72, 72

Offset: 0

W. Edwin Clark, Jul 27 2021

nonn

This agrees with A052928 until n = 2436. a(2436) = 2434 and A052928(2436) = 2436. For n >= 2436 we have a(n) = n-2 for n even and a(n) = n - 1 for n odd, for an as yet undetermined number of n. A052928(n) = n - (n mod 2) for all n. See the paper by W. Clark and M. Shattuck linked to below.

W. Edwin Clark and Mark Shattuck, The Integer Sequence Transform a --> b where b_n is the Number of Real Roots of the Polynomial a_0 + a_1 x + a_2 x^2 + ... + a_n x^n, arXiv:2107.05572 [math.CO], 2021.
Index entries for sequences which agree for a long time but are different

Cf. A052928, A346442, A346443, A346444, A346445.

cp's OEIS Frontend

User: W. Edwin Clark

A376691 The number of solutions to x + y == x^2 + y^2 == x^3 + y^3 (mod n) with 0 <= x <= y < n.

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A376646 Number of solutions to x + y == x^2 + y^2 (mod n) with x <= y.

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Maple

PARI

Python

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A376318 The number of distinct values of x+y+z (mod n) when x*y*z = 1 (mod n).

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Maple

PARI

Python

A376427 The number of distinct values of x+y+z+w (mod n) when x*y*z*w = 1 (mod n).

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Maple

Python

A376296 The number of solutions x<=y<=z<=w in Z/(n) of the equation x+y+z+w = x*y*z*w.

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Maple

Python

A376183 The number of solutions x<=y<=z in Z/(n) of the equation x+y+z = x*y*z.

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Programs

Maple

PARI

Python

Extensions

A375625 Number of distinct values taken by F(p) = Sum (|i-j| - |p(i)-p(j)|)^2 where the sum is over all 1 <= i < j <= n, for all permutations p in the symmetric group S_n.

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Author

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Programs

Maple

Extensions

A375623 Maximum value of F(p) = Sum (|i-j| - |p(i)-p(j)|)^2 where the sum is over all 1 <= i < j <= n, for all permutations p in the symmetric group S_n.

A376318 The number of distinct values of x+y+z (mod n) when xyz = 1 (mod n).

A376427 The number of distinct values of x+y+z+w (mod n) when xyz*w = 1 (mod n).

A376296 The number of solutions x<=y<=z<=w in Z/(n) of the equation x+y+z+w = xyz*w.

A376183 The number of solutions x<=y<=z in Z/(n) of the equation x+y+z = xyz.