Bence Bernáth - cp's OEIS frontend

A385713 Numbers w + x + y + z where (w,x,y,z) are nonnegative solutions to the Diophantine equation w^2 + x^2 + y^2 + z^2 = wxy*z.

0, 8, 12, 32, 108, 292, 392, 1452, 3392, 3712, 5408, 11808, 20172, 34872, 40332, 50572, 75272, 162212, 280908, 480512, 518252, 595052, 700352, 1048352, 1639308, 3912492, 4599172, 4911172, 5725732, 6049292, 6508932, 7132608, 8279808, 9739812

Offset: 1

Bence Bernáth, Jul 07 2025

nonn

The trivial sum with x=y=z=w=0 is included. The solutions are generated by the Vieta jumping discussed in the Numberphile video. The algorithm finds all the quadruplet solutions within the radius of a 4D sphere. Then the quadruplets are sorted based on their sum in increasing order.

			8 is a term because (w,x,y,z) = (2,2,2,2) is the first nontrivial solution.
12 is a term because (2,2,2,6) is a solution.

Bence Bernáth, Table of n, a(n) for n = 1..10000
Michael Magee and Brady Haran, A simple equation that behaves weirdly, Numberphile, Youtube video, 2025.

Cf. A061292, A380462.

import math
def is_perfect_square(n):
    return (math.isqrt(n)) ** 2 == n
def generate_all_solutions(radius):
    solutions = set()
    visited = set()
    # Initial known solution
    seed = (2, 2, 2, 2)
    queue = [seed]
    solutions.add(seed)
    while queue:
        quad = queue.pop(0)
        for ii in range(4):
            # Cyclically rotate variables: solve for the ii-th one
            rotated = list(quad[ii:] + quad[:ii])
            x, y, z, w = rotated
            # Use Vieta's formula: x^2 - (yzw)x + (y² + z² + w²) = 0
            product = y * z * w
            sumsq = y**2 + z**2 + w**2
            D = product**2 - 4 * sumsq
            if D < 0 or not is_perfect_square(D):
                continue
            sqrt_D = (math.isqrt(D))
            for sign in [+1, -1]:
                x_new = (product + sign * sqrt_D)
                if x_new % 2 != 0:
                    continue
                x_new //= 2
                if not (0 < x_new < radius):
                    continue
                new_quad = [x_new, y, z, w]
                if any(val >= radius for val in new_quad):
                    continue
                new_quad_sorted = tuple(sorted(new_quad))
                if new_quad_sorted not in visited:
                    solutions.add(new_quad_sorted)
                    queue.append(tuple(new_quad))
                    visited.add(new_quad_sorted)
    # Return solutions sorted by sum of elements
    return sorted([list(sol) for sol in solutions], key=lambda tup: sum(tup))
print([sum(sol) for sol in generate_all_solutions(10000000)])

A380462 a(n) is the number of positive solutions to the Diophantine equation w^2 + x^2 + y^2 + z^2 = wxy*z such that x,y,z,w < e^n.

Original entry on oeis.org

1, 2, 2, 3, 4, 6, 6, 7, 10, 12, 16, 17, 18, 23, 25, 32, 35, 39, 43, 47, 55, 60, 64, 73, 76, 86, 94, 101, 111, 118, 132, 141, 150, 159, 168, 180, 192, 203, 220, 235, 247, 261, 275, 289, 302, 324, 340, 361, 376, 394, 413, 433, 454, 472, 498, 518, 536, 560, 584, 606, 632, 658, 684

Offset: 1

Bence Bernáth, Jun 22 2025

nonn

The trivial x=y=z=w=0 solution is not included. The solutions are generated by the Vieta jumping discussed in the Numberphile video.

			a(3)=2 because the solutions below e^3=20.085... are [2,2,2,2] and [2,2,2,6]. One quadruplet solution is counted only once.

Bence Bernáth, Table of n, a(n) for n = 1..650
Michael Magee and Brady Haran, A simple equation that behaves weirdly, Numberphile, Youtube video, 2025.

Cf. A061292.

import math
from collections import deque
def is_perfect_square(n):
    return (math.isqrt(n)) ** 2 == n
def generate_all_solutions(up_to_bound):
    solutions = set()
    visited = set()
    seed = (2, 2, 2, 2)
    queue = deque([seed])
    solutions.add(seed)
    while queue:
        quad = queue.popleft()
        for ii in range(4):
            rotated = list(quad[ii:] + quad[:ii])
            x, y, z, w = rotated
            product = y * z * w
            sumsq = y**2 + z**2 + w**2
            D = product**2 - 4 * sumsq
            if D < 0 or not is_perfect_square(D):
                continue
            sqrt_D = (math.isqrt(D))
            for sign in [+1, -1]:
                x_new = (product + sign * sqrt_D)
                if x_new % 2 != 0:
                    continue
                x_new //= 2
                if not (0 < x_new < up_to_bound):
                    continue
                new_quad = [x_new, y, z, w]
                if any(val >= up_to_bound for val in new_quad):
                    continue
                new_quad_sorted = tuple(sorted(new_quad))
                if new_quad_sorted not in visited:
                    solutions.add(new_quad_sorted)
                    queue.append(tuple(new_quad))
                    visited.add(new_quad_sorted)
    return sorted(solutions)
# Compute all solutions up to e^100
max_bound = math.exp(100)
all_solutions = generate_all_solutions(max_bound)
# Precompute max entry of each solution for fast thresholding
solution_maxes = [max(sol) for sol in all_solutions]
# Generate the sequence a(n) for n = 1 to 100
a_n = []
for n in range(1, 101):
    threshold = math.exp(n)
    count = sum(1 for m in solution_maxes if m < threshold)
    a_n.append(count)
print(a_n)

A369104 Decimal expansion of Pi*(Pi-2)/(4+(Pi-2)^2).

Original entry on oeis.org

6, 7, 6, 2, 7, 0, 2, 2, 2, 6, 8, 6, 3, 4, 5, 1, 5, 7, 4, 1, 2, 6, 3, 1, 0, 5, 3, 5, 7, 4, 5, 0, 8, 0, 5, 0, 9, 0, 2, 1, 6, 6, 8, 4, 5, 8, 1, 3, 0, 7, 0, 7, 2, 2, 2, 5, 3, 1, 9, 8, 6, 1, 7, 4, 1, 6, 1, 2, 4, 8, 6, 7, 8, 3, 8, 8, 4, 9, 4, 6, 7, 3, 4, 0, 6, 5, 8, 5, 7, 4, 0, 6, 6, 2, 3, 0, 3, 9, 7, 9, 0, 0, 4, 1, 7, 5, 8, 7, 9, 4, 5, 8, 2, 4, 8, 3, 8, 6, 5, 5, 5, 2, 2, 6, 0, 8, 2, 7, 9, 5, 4, 6, 6, 7, 7

Offset: 0

Bence Bernáth, Jan 13 2024

This is the prefactor of the Hall resistivity of strange metals in the high magnetic field regime. The number comes from a phenomenological model using a modified Boltzmann equation. See the link for details.

			0.67627022268...

R. D. H. Hinlopen, F. A. Hinlopen, J. Ayres, and N. E. Hussey, B^2 to B-linear magnetoresistance due to impeded orbital motion, Physical Review Research 4, no. 3 (2022): 033195.

Cf. A369103, A019669, A000796.

First[RealDigits[Pi*(Pi-2)/(4+(Pi-2)^2), 10, 120, 0]] (* Paolo Xausa, Jun 29 2024 *)

from sympy import pi, N;
def A369104(n):
    return list(map(int, str(N((pi*(pi-2))/(4+(pi-2)**2), n)).replace(".", "")))

Leading 0 removed and offset changed to 0 by Georg Fischer, Feb 03 2025

A369103 Decimal expansion of 2*Pi/(4+(Pi-2)^2).

Original entry on oeis.org

1, 1, 8, 4, 7, 8, 3, 7, 6, 7, 7, 6, 4, 7, 6, 4, 0, 3, 9, 3, 1, 2, 1, 1, 1, 3, 8, 5, 2, 5, 4, 1, 9, 3, 0, 9, 2, 3, 7, 1, 1, 5, 1, 2, 3, 9, 3, 6, 0, 4, 4, 8, 9, 6, 2, 9, 3, 2, 7, 6, 2, 2, 9, 0, 3, 8, 3, 0, 7, 6, 6, 4, 4, 1, 3, 8, 1, 5, 1, 6, 8, 2, 7, 3, 3, 9, 8, 6, 8, 3, 7, 3, 8, 2, 7, 9, 5

Offset: 1

Bence Bernáth, Jan 13 2024

This is the prefactor of the linear magnetoresistance of strange metals in the high magnetic field regime. The number comes from a phenomenological model using a modified Boltzmann equation. See the link for details.

			1.18478376776...

R. D. H. Hinlopen, F. A. Hinlopen, J. Ayres, and N. E. Hussey, B^2 to B-linear magnetoresistance due to impeded orbital motion, Physical Review Research 4, no. 3 (2022): 033195.

Cf. A369104, A019669, A000796.

First[RealDigits[2 Pi/(4+(Pi-2)^2), 10, 120]] (* Paolo Xausa, Jan 13 2024 *)

from sympy import pi, N;
def A369103(n):
    return list(map(int, str(N(2*pi/(4+(pi-2)**2), n)).replace(".", "")))

A359031 a(n+1) gives the number of occurrences of the mode of the digits of a(n) among all the digits of [a(0), a(1), ..., a(n)], with a(0)=0.

Original entry on oeis.org

0, 1, 1, 2, 1, 3, 1, 4, 1, 5, 1, 6, 1, 7, 1, 8, 1, 9, 1, 10, 2, 2, 3, 2, 4, 2, 5, 2, 6, 2, 7, 2, 8, 2, 9, 2, 10, 3, 3, 4, 3, 5, 3, 6, 3, 7, 3, 8, 3, 9, 3, 10, 4, 4, 5, 4, 6, 4, 7, 4, 8, 4, 9, 4, 10, 5, 5, 6, 5, 7, 5, 8, 5, 9, 5, 10, 6, 6, 7, 6, 8, 6, 9, 6, 10, 7, 7, 8, 7, 9, 7, 10

Offset: 0

Bence Bernáth, Dec 12 2022

The mode is the most frequently occurring value among the digits of a(n). When there are multiple values occurring equally frequently, the mode is the smallest of those values.

Up to a(464)=110, the terms are identical to A358967.

Bence Bernáth, Table of n, a(n) for n = 0..20000
Eric Angelini, Digit-counters updating themselves

Cf. A248034, A249009, A356348, A336514, A358967, A358851, A322182.

length_seq=470;
sequence(1)=0;
seq_for_digits=(num2str(sequence(1))-'0');
for i1=1:1:length_seq
     sequence(i1+1)=sum(seq_for_digits==mode((num2str(sequence(i1))-'0'))');
     seq_for_digits=[seq_for_digits, num2str(sequence(i1+1))-'0'];
end

import statistics as stat
sequence=[0]
length=470
seq_for_digits=list(map(int, list(str(sequence[0]))))
for ii in range(length):
    sequence.append(seq_for_digits.count(stat.mode(list(map(int, list(str(sequence[-1])))))))
    seq_for_digits.extend(list(map(int, list(str(sequence[-1])))))

A358851 a(n+1) is the number of occurrences of the largest digit of a(n) among all the digits of [a(0), a(1), ..., a(n)], with a(0)=0.

Original entry on oeis.org

0, 1, 1, 2, 1, 3, 1, 4, 1, 5, 1, 6, 1, 7, 1, 8, 1, 9, 1, 10, 11, 13, 2, 2, 3, 3, 4, 2, 4, 3, 5, 2, 5, 3, 6, 2, 6, 3, 7, 2, 7, 3, 8, 2, 8, 3, 9, 2, 9, 3, 10, 15, 4, 4, 5, 5, 6, 4, 6, 5, 7, 4, 7, 5, 8, 4, 8, 5, 9, 4, 9, 5, 10, 17, 6, 6, 7, 7, 8, 6

Offset: 0

Bence Bernáth, Dec 08 2022

Up to a(19)=10, the terms are identical to A248034. The branches (distinct lines of terms indicating the largest digit of the preceding term) can be labeled by the counter digit (shown in the scatter plot). From 9 to 1 the branches gradually get fragmented. Below digit 5 it is harder to disentangle the branches in some places. A repeating pattern also appears (shown in the inset of the scatter plot).

Bence Bernáth, Table of n, a(n) for n = 0..10000
Eric Angelini, Digit-counters updating themselves
Bence Bernáth, Scatter plot for n=0..500000
Michael De Vlieger, Log log scatterplot of a(n) n = 1..10^6.
Michael De Vlieger, Log log scatterplot of a(n), n = 1..10^5, with a color function indicating largest digit of preceding term, where black = 0, red = 1, orange = 2, ..., magenta = 9.

Cf. A248034, A249009, A356348, A336514.

length_seq=150;
sequence(1)=0;
seq_for_digits=(num2str(sequence(1))-'0');
for i1=1:1:length_seq
     sequence(i1+1)=sum(seq_for_digits==max((num2str(sequence(i1))-'0'))');
     seq_for_digits=[seq_for_digits, num2str(sequence(i1+1))-'0'];
end

nn = 79; s[] = 0; a[0] = 0; Do[(Set[d, Max[#]]; Map[s[#1] += #2 & @@ # &, Tally[#] ]) &@ IntegerDigits[a[n - 1]]; a[n] = s[d], {n, nn}]; Array[a, nn + 1, 0] (* _Michael De Vlieger, Dec 28 2022 *)

sequence=[0]
length=150
seq_for_digits=list(map(int, list(str(sequence[0]))))
for ii in range(length):
     sequence.append(seq_for_digits.count(max(list(map(int,list(str(sequence[-1])))))))
     seq_for_digits.extend(list(map(int, list(str(sequence[-1])))))

A358967 a(n+1) gives the number of occurrences of the smallest digit of a(n) so far, up to and including a(n), with a(0)=0.

Original entry on oeis.org

0, 1, 1, 2, 1, 3, 1, 4, 1, 5, 1, 6, 1, 7, 1, 8, 1, 9, 1, 10, 2, 2, 3, 2, 4, 2, 5, 2, 6, 2, 7, 2, 8, 2, 9, 2, 10, 3, 3, 4, 3, 5, 3, 6, 3, 7, 3, 8, 3, 9, 3, 10, 4, 4, 5, 4, 6, 4, 7, 4, 8, 4, 9, 4, 10, 5, 5, 6, 5, 7, 5, 8, 5, 9, 5, 10, 6, 6, 7, 6, 8, 6, 9, 6, 10, 7, 7, 8, 7, 9, 7, 10, 8, 8, 9, 8, 10, 9, 9, 10, 10, 11, 22, 12, 23, 14

Offset: 0

Bence Bernáth, Dec 08 2022

Up to a(103)=12, the terms are identical to A248034.

Bence Bernáth, Table of n, a(n) for n = 0..10000
Eric Angelini, Digit-counters updating themselves

Cf. A248034, A249009, A356348, A336514, A358851.

length_seq=150;
sequence(1)=0;
seq_for_digits=(num2str(sequence(1))-'0');
for i1=1:1:length_seq
     sequence(i1+1)=sum(seq_for_digits==min((num2str(sequence(i1))-'0'))');
     seq_for_digits=[seq_for_digits, num2str(sequence(i1+1))-'0'];
end

sequence=[0]
length=150
seq_for_digits=list(map(int, list(str(sequence[0]))))
for ii in range(length):
   sequence.append(seq_for_digits.count(min(list(map(int,list(str(sequence[-1])))))))
   seq_for_digits.extend(list(map(int, list(str(sequence[-1])))))

A358520 Nearest integer to n/sin(n).

Original entry on oeis.org

1, 2, 21, -5, -5, -21, 11, 8, 22, -18, -11, -22, 31, 14, 23, -56, -18, -24, 127, 22, 25, -2486, -27, -27, -189, 34, 28, 103, -44, -30, -77, 58, 33, 64, -82, -36, -57, 128, 40, 54, -258, -46, -52, 2486, 53, 51, 380, -62, -51, -191, 76, 53, 134, -97

Offset: 1

Bence Bernáth, Nov 20 2022

It is also the nearest integer to sinc(x)^(-1) function.

			For n=3, 3/sin(3) = 21.25..., therefore a(3) = 21.

Bence Bernáth, Table of n, a(n) for n = 1..10000
Wikipedia, Sinc function

Cf. A272695, A051422, A105666.

Cf. A046947 (Values for n where abs(a(n))/n has records).

Mathematica
```
Table[Round[n/Sin[n]], {n, 1, 100}]
```

a(n) = round(n/sin(n)); \\ Michel Marcus, Nov 20 2022

A337014 a(1) = 0 and for n > 1, a(n+1) = (k(n) - a(n))*(-1)^(n+1) where k(n) is the number of terms equal to a(n) among the first n terms.

Original entry on oeis.org

0, 1, 0, 2, 1, 1, -2, 3, 2, 0, -3, 4, 3, -1, -2, 4, 2, 1, -3, 5, 4, -1, -3, 6, 5, -3, -7, 8, 7, -6, -7, 9, 8, -6, -8, 9, 7, -5, -6, 9, 6, -4, -5, 7, 4, 0, -4, 6, 3, 0, -5, 8, 5, -2, -5, 9, 5, -1, -4, 7, 3, 1, -4, 8, 4, 1, -5, 10, 9, -4, -9, 10, 8, -3, -8, 10, 7, -2, -6, 10, 6, -2, -7

Offset: 1

Bence Bernáth, Nov 21 2020

The graph of the sequence shows interesting, "Christmas tree"-like shapes.

			For n = 4, a(4) = 2, which appeared only once before, so a(5)=(1-2)*(-1)^5 = 1.

Bence Bernáth, Table of n, a(n) for n = 1..10000

Cf. A337835, A329985.

length=10000;
sequence(1)=0;
for n=2:1:length
     sequence(n)=((nnz(sequence==sequence(end)))-(sequence(n-1)))*(-1)^n;
end

a = {0}; Do[AppendTo[a, (-1)^k*(Count[a, a[[-1]]] - a[[-1]])], {k, 0, 81}]; a (* Amiram Eldar, Nov 21 2020 *)

{ for (n=1, #a=vector(83), print1(a[n]=if (n==1, 0, k=sum(k=1, n-1, a[k]==a[n-1]); (k-a[n-1])*(-1)^n) ", ")) } \\ Rémy Sigrist, Nov 22 2020

from itertools import islice
from collections import Counter
def agen(): # generator of terms
    an, k, sign = 0, Counter(), -1
    while True:
        yield an
        k[an] += 1
        sign *= -1
        an = (k[an] - an)*sign
print(list(islice(agen(), 83))) # Michael S. Branicky, Nov 12 2022

A337835 a(1) = 0 and for n > 1, a(n+1) = k - a(n) where k is the number of terms equal to a(n) among the first n terms.

Original entry on oeis.org

0, 1, 0, 2, -1, 2, 0, 3, -2, 3, -1, 3, 0, 4, -3, 4, -2, 4, -1, 4, 0, 5, -4, 5, -3, 5, -2, 5, -1, 5, 0, 6, -5, 6, -4, 6, -3, 6, -2, 6, -1, 6, 0, 7, -6, 7, -5, 7, -4, 7, -3, 7, -2, 7, -1, 7, 0, 8, -7, 8, -6, 8, -5, 8, -4, 8, -3, 8, -2, 8, -1, 8, 0, 9, -8, 9, -7, 9, -6, 9, -5, 9, -4, 9, -3, 9

Offset: 1

Bence Bernáth, Nov 17 2020

Empirical observation: -A000194(n) <= a(n) <= A000194(n).

			For n = 4, a(4) = 2, which appeared only once in the sequence so a(5)=1-2=-1.

Bence Bernáth, Table of n, a(n) for n = 1..10000

Cf. A000194, A329985.

length=1000;
sequence(1)=0;
for n=2:1:length
        sequence(n)=nnz(sequence==sequence(end))-sequence(n-1);
end

a = {0}; Do[AppendTo[a, Count[a, a[[-1]]] - a[[-1]]], {100}]; a (* Amiram Eldar, Nov 18 2020 *)
a[1] = 0; a[n_Integer?Positive] := a[n] = Count[Array[a, n - 1], a[n - 1]] - a[n - 1]; Array[a, 101] (* Jan Mangaldan, Nov 27 2020 *)

lista(nn) = {my(va = vector(nn)); for (n=2, nn, va[n] = #select(x->(x==va[n-1]), vector(n-1, k, va[k])) - va[n-1];); va;} \\ Michel Marcus, Nov 18 2020

from itertools import islice
from collections import Counter
def agen(): # generator of terms
    an, k = 0, Counter()
    while True:
        yield an
        k[an] += 1
        an = k[an] - an
print(list(islice(agen(), 86))) # Michael S. Branicky, Nov 12 2022

cp's OEIS Frontend

User: Bence Bernáth

A385713 Numbers w + x + y + z where (w,x,y,z) are nonnegative solutions to the Diophantine equation w^2 + x^2 + y^2 + z^2 = w*x*y*z.

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A380462 a(n) is the number of positive solutions to the Diophantine equation w^2 + x^2 + y^2 + z^2 = w*x*y*z such that x,y,z,w < e^n.

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A369104 Decimal expansion of Pi*(Pi-2)/(4+(Pi-2)^2).

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A369103 Decimal expansion of 2*Pi/(4+(Pi-2)^2).

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A359031 a(n+1) gives the number of occurrences of the mode of the digits of a(n) among all the digits of [a(0), a(1), ..., a(n)], with a(0)=0.

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A358851 a(n+1) is the number of occurrences of the largest digit of a(n) among all the digits of [a(0), a(1), ..., a(n)], with a(0)=0.

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A358967 a(n+1) gives the number of occurrences of the smallest digit of a(n) so far, up to and including a(n), with a(0)=0.

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A358520 Nearest integer to n/sin(n).

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A385713 Numbers w + x + y + z where (w,x,y,z) are nonnegative solutions to the Diophantine equation w^2 + x^2 + y^2 + z^2 = wxy*z.

A380462 a(n) is the number of positive solutions to the Diophantine equation w^2 + x^2 + y^2 + z^2 = wxy*z such that x,y,z,w < e^n.