Zachary DeStefano - cp's OEIS frontend

A370279 Indices of record high values in A370278.

2, 3, 23, 118, 217, 611, 1299, 1949, 3331, 4403

Offset: 1

Zachary DeStefano, Feb 13 2024

			The 118th index of A370278 is the first occurrence of the value 3 in that sequence. All prior values are 0, 1, or 2.

Cf. A370277, A370278.

A370278 Difference between the bound provided by Dirichlet's Simultaneous Approximation Theorem applied to Z_n (for d=3) and the best possible bound.

Original entry on oeis.org

0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 1, 2, 2, 1, 0, 0, 1, 1, 0, 0, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 2, 2, 2, 1, 1, 2, 2, 2, 1, 0, 0, 2, 1, 1, 0, 0, 0, 1, 1, 1, 1, 1, 0, 2, 2, 1, 2, 1, 1, 2, 2, 2, 1, 2, 1, 1, 2, 2, 2, 2, 1, 1, 1, 2, 2, 2, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3

Offset: 2

Zachary DeStefano, Feb 13 2024

nonn

Indices where this sequence is 0 form the sequence A370277.

The indices of record high values form the sequence A370279.

			For n = 6, floor(k^(2/3)) = 3, but for all triples (a_1, a_2, a_3), there is a choice of p such that |p*a_1| mod 6, |p*a_2| mod 6, and |p*a_3| mod 6 are all smaller than or equal to 2.
For example, consider the triple (1, 2, 3), with p = 2; we have:
|2 * 1| mod 6 = 2, |2 * 2| mod 6 = 2, and |2 * 3| mod 6 = 0.
Note that there is no nonzero choice of p such that all values are smaller than 2 for this triple.

Cf. A370277, A370279.

A370277 Numbers k with the property that Dirichlet's Simultaneous Approximation Theorem applied to Z_k is tight (for d = 3).

Original entry on oeis.org

2, 4, 5, 7, 8, 10, 11, 14, 18, 26, 27, 30, 31, 63, 64, 68, 69, 70, 76, 124, 125, 130, 131, 132, 148, 215, 216, 222, 223, 224, 225, 234, 342, 343, 350, 351, 352, 353

Offset: 1

Zachary DeStefano, Feb 13 2024

Dirichlet's Simultaneous Approximation Theorem applied to Z_k states that for all a_1, a_2, ..., a_d, there exists a nonzero p such that |pa_i| <= k^(1 - 1/d) mod k.
For d = 3, the bound of floor(k^(2/3)) is tight only for specific values of k. That is to say, max_(a_1,a_2,a_3) min_p max_i |pa_i| = floor(k^(2/3)) only for specific values of k. These are those values.
This sequence consists of the indices of the zeros in A370278.
It appears that this sequence contains all integers k such that k or k+1 is a cube.

			For k = 14, floor(k^(2/3)) = 5. Given the triple (1, 3, 5), there is no choice of p such that |p| mod 14, |3p| mod 14, and |5p| mod 14 are all smaller than 5.
p = 1, 3, 5, 9, 11, and 13 results in a simultaneous minimum of 5.

Stack Exchange Dirichlet's Simultaneous Approximation Bound.

Cf. A370278, A370279.

A363069 Size of the largest subset of {1,2,...,n} such that no two elements sum to a perfect square.

Original entry on oeis.org

1, 1, 1, 2, 2, 3, 4, 4, 4, 4, 5, 5, 6, 6, 6, 7, 8, 8, 8, 8, 9, 9, 10, 10, 11, 11, 12, 12, 12, 13, 13, 13, 13, 14, 14, 14, 15, 15, 16, 16, 17, 17, 18, 18, 18, 19, 19, 19, 20, 20, 20, 20, 21, 21, 22, 22, 23, 23, 24, 24, 25, 25, 25, 25, 26, 26, 26, 26, 26, 27, 27

Offset: 1

Zachary DeStefano, May 16 2023

nonn

			The first few examples where a(n) increases are {1}, {1,4}, {1,4,6}, and {1,4,6,7}.

Z. DeStefano, Maximum Sized Sets With Sums That Avoid Squares

Cf. A100719, A210380.

The set: {k | k <= n, k == 1 (mod 3)} provides a lower bound: a(n) >= floor((n+2)/3).

A361340 a(n) = smallest number with the property that the split-and-multiply technique (see A361338) in base n can produce all n single-digit numbers.

Original entry on oeis.org

15, 23, 119, 167, 12049, 424, 735, 907, 17117, 1250, 307747, 2703, 49225, 9422, 57823, 5437, 2076131, 7747, 639987, 44960, 822799, 11537, 23809465, 24967, 1539917, 109346, 4643181, 26357, 5587832443, 37440, 1885949, 285085, 7782015, 265806, 1250473675, 66524, 8340541, 699890, 158607997, 85684

Offset: 2

N. J. A. Sloane, Apr 04 2023, based on an email from Zachary DeStefano

From Zachary DeStefano, May 17 2023: (Start)

There is a strong linear relationship between n^(n / phi(n)) and a(n) (see A000010 for phi(n)) which results from the final digit falling into subgroups of Z/nZ during split-and-multiply steps. This explains why a(n) is significantly smaller for prime n and significantly larger when n contains several small prime factors (ex. 2 * 3 * 5 = 30) (End)

			To reach the digits 0 though 9 in base 10 from 17117:
 171*17 -> 290*7  -> 203*0 -> 0
 1711*7 -> 1197*7 -> 837*9 -> 7*533 -> 373*1 -> 37*3  -> 1*11 -> 1*1 -> 1
 171*17 -> 2*907  -> 1*814 -> 8*14  -> 1*12  -> 1*2   -> 2
 1*7117 -> 711*7  -> 49*77 -> 377*3 -> 113*1 -> 1*13  -> 1*3  -> 3
 171*17 -> 2*907  -> 1*814 -> 8*14  -> 11*2  -> 2*2   -> 4
 1711*7 -> 1197*7 -> 837*9 -> 75*33 -> 247*5 -> 1*235 -> 23*5 -> 1*15 -> 1*5  -> 5
 17*117 -> 19*89  -> 169*1 -> 16*9  -> 1*44  -> 4*4   -> 1*6  -> 6
 1711*7 -> 1197*7 -> 837*9 -> 7*533 -> 37*31 -> 11*47 -> 51*7 -> 3*57 -> 17*1 -> 1*7 -> 7
 17*117 -> 1*989  -> 98*9  -> 88*2  -> 1*76  -> 7*6   -> 4*2  -> 8
 1*7117 -> 711*7  -> 49*77 -> 377*3 -> 113*1 -> 11*3  -> 3*3  -> 9

Zachary DeStefano and Tim Peters, Table of n, a(n) for n = 2..119
Michael S. Branicky, Python program
Michael De Vlieger, Plot of digit d in sequence S_n(m) in black at (x, y) = (m, d) for bases n = 2..12 as labeled, m = 1..1000, and digits d = 0..n-1, 12X exaggeration. This sequence shows the first occasion of a vertical black bar in base n.
Zachary DeStefano and Tim Peters, Known terms and bounds

Cf. A361337-A361349.

Table[Catch[Monitor[Do[(Set[c, Count[Union@Flatten[#], ?(# < b &)]]; If[c == b, Throw[i]]) &@ NestWhileList[Flatten@ Map[Function[w, Array[If[And[#[[-1, 1]] == 0, Length[#[[-1]]] > 1], Nothing, Times @@ Map[FromDigits[#, b] &, #]] &@ TakeDrop[w, #] &, Length[w] - 1]][IntegerDigits[#, b]] &, #] &, {i}, Length[#] > 0 &], {i, 0, Infinity}], {b, i, c}]], {b, 2, 6}] (* _Michael De Vlieger, Apr 04 2023, with Monitor to show progress *)

from itertools import count
from sympy.ntheory import digits
from functools import lru_cache
def fd(d, b): # from_digits
    return sum(di*b**i for i, di in enumerate(d[::-1]))
@lru_cache(maxsize=None)
def f(n, b):
    if n < b: return {n}
    s = digits(n, b)[1:]
    return {e for i in range(1, len(s)) if s[i]!=0 or i==len(s)-1 for e in f(fd(s[:i], b)*fd(s[i:], b), b)}
def a(n, printat=False):
    return next(k for k in count(1) if len(f(k, n))==n)
print([a(n) for n in range(2, 18)]) # Michael S. Branicky, Apr 04 2023

# see link for a version that is faster and uses less memory

a(21)-a(29) from Michael S. Branicky, Apr 04 2023

a(30)-a(41) from Zachary DeStefano, Apr 05 2023

A358177 Number of Eulerian orientations of a (labeled) 2n-dimensional hypercube graph, Q_2n. Q_2n is also the n-dimensional torus grid graph (C_4)^n.

Original entry on oeis.org

1, 2, 2970, 351135773356461511142023680

Offset: 0

Peter Munn and Zachary DeStefano, Nov 02 2022

An Eulerian orientation of a graph is an orientation of the edges such that every vertex has in-degree equal to out-degree. (C_4)^n denotes the Cartesian product of n cycle graphs on 4 nodes.

			For n = 1, dimension 2n = 2, there are two Eulerian orientations (the cyclic ones). So a(1) = 2.

Mikhail Isaev, Brendan D. McKay, and Rui-Ray Zhang, Correlation between residual entropy and spanning tree entropy of ice-type models on graphs, arXiv:2409.04989 [math.CO], 2024. See p. 24.
A. Schrijver, Bounds on the Number of Eulerian Orientations, Combinatorica, 3 (3--4), (1983), 375-380.
Eric Weisstein's World of Mathematics, Cartesian product of graphs
Eric Weisstein's World of Mathematics, Hypercube Graph

Cf. A007081, A054759, A298119, A307334, A334553.

a(0) = A007081(2^0) = 1.
a(1) = A334553(1) = 2.
a(2) = A054759(4) = 2970.
Schrijver (1983) provides general bounds on unknown terms of the form (2^(-k) * binomial(2k,k))^(2^(2k)) <= a(k) <= sqrt(binomial(2k,k)^(2^(2k))).
From this we have the specific bounds 2.9*10^25 <= a(3) <= 4.3*10^41 and 1.2*10^164 <= a(4) <= 1.5*10^236.

a(3) added by Brendan McKay, Nov 04 2022

A350547 Maximum size of a set of points taken from a hexagonal section of a hexagonal grid with side length n such that no three selected points form an equilateral triangle.

Original entry on oeis.org

1, 4, 9, 15, 22, 28, 36

Offset: 0

Zachary DeStefano, Jan 06 2022

The hexagon with side length n has n+1 points along each edge and contains a total of A003215(n) points.
The following lower bounds are known:
a(7) >= 44;
a(8) >= 52;
a(9) >= 60.
All currently known values and lower bounds can be achieved by a configuration with reflective symmetry.

			For n = 4 the a(4) = 22 solution, unique up to rotation, is:
.
      o x x o x
     x x o o o x
    x o o o o x o
   o o o o o o x x
  x o o o o o o o x
   x x o o o o o o
    o x o o o o x
     x o o o x x
      x o x x o
.

Cf. A003215, A008893, A240114.

A343828 Numbers which are the product of two S-primes (A057948) in exactly three ways.

Original entry on oeis.org

4389, 5313, 7161, 9177, 9933, 10857, 12369, 13629, 14421, 14973, 15477, 16401, 17157, 18249, 18753, 19173, 19437, 20769, 22701, 23529, 23541, 23793, 24717, 26733, 26961, 27993, 28329, 28497, 29337, 29469, 30261, 30597, 31521, 32109, 32361, 32637, 33117, 33649

Offset: 1

Zachary DeStefano, Apr 30 2021

nonn

There exist numbers which are the product of two S-primes in exactly 1, 2, and 3 ways.

An S-prime is either a prime of the form 4k+1 or a semiprime of the form (4k+3)*(4m+3). That means the maximum number of prime factors that a number factorizable into two S-primes can have is four (all 4k + 3), and those can be combined into S-primes in at most three distinct ways. - Gleb Ivanov, Dec 07 2021

			9177 = 21*437 = 57*161 = 69*133 which are all S-primes (A057948), and admits no other S-Prime factorizations.
4389 = (3*7)*(11*19) = (3*11)*(7*19) = (3*19)*(7*11); 3,7,11,19 are the smallest primes of the form 4k + 3.

Zachary DeStefano, Table of n, a(n) for n = 1..1014

Cf. A054520, A057948, A057949, A057950.

Exactly one way: A343826. Exactly two ways: A343827.

\\ uses is(n) from A057948
isok(n) = sumdiv(n, d, (d<=n/d) && is(d) && is(n/d)) == 3; \\ Michel Marcus, May 01 2021

a(n) == 1 (mod 4). - Hugo Pfoertner, May 01 2021

A343827 Numbers which are the product of two S-primes (A057948) in exactly two ways.

Original entry on oeis.org

441, 693, 1089, 1197, 1449, 1617, 1881, 1953, 2277, 2541, 2709, 2793, 2961, 3069, 3249, 3381, 3717, 3933, 4221, 4257, 4473, 4557, 4653, 4761, 4977, 5229, 5301, 5841, 5929, 6321, 6417, 6489, 6633, 6741, 6897, 6909, 7029, 7353, 7581, 7821, 8001, 8037, 8217, 8253

Offset: 1

Zachary DeStefano, Apr 30 2021

nonn

First differs from A057950 at a(21)=4473, whereas A057950(21)=4389, which can be represented as the product of two S-primes in exactly 3 ways.

There exist numbers which are the product of two S-primes in exactly 1, 2, and 3 ways; however, it is unknown if any numbers exist which are the product of two S-primes in exactly 4 ways.

			1449=9*161=21*69 which are all S-primes (A057948), and admits no other S-prime factorizations.

Zachary DeStefano, Table of n, a(n) for n = 1..2484

Cf. A054520, A057948, A057949, A057950.

Exactly one way: A343826. Exactly three ways: A343828.

\\ uses is(n) from A057948
isok(n) = sumdiv(n, d, (d<=n/d) && is(d) && is(n/d)) == 2; \\ Michel Marcus, May 01 2021

a(n) == 1 (mod 4). - Hugo Pfoertner, May 01 2021

A343826 Numbers which are the product of two S-primes (A057948) in exactly one way.

Original entry on oeis.org

25, 45, 65, 81, 85, 105, 117, 145, 153, 165, 169, 185, 189, 205, 221, 245, 261, 265, 273, 285, 289, 297, 305, 333, 345, 357, 365, 369, 377, 385, 429, 445, 465, 477, 481, 485, 493, 505, 513, 533, 545, 549, 561, 565, 605, 609, 621, 629, 637, 645, 657, 665, 685

Offset: 1

Zachary DeStefano, Apr 30 2021

nonn

There exist numbers which are the product of two S-primes in exactly 1, 2, and 3 ways; however, it is unknown if any numbers exist which are the product of two S-primes in exactly 4 ways.

			153 = 9*17 which are both S-primes, and admits no other S-prime factorizations.

Zachary DeStefano, Table of n, a(n) for n = 1..3786

Cf. A054520, A057948, A057949, A057950.

Exactly two ways: A343827. Exactly three ways: A343828.

\\ uses is(n) from A057948
isok(n) = sumdiv(n, d, (d<=n/d) && is(d) && is(n/d)) == 1; \\ Michel Marcus, May 01 2021

a(n) == 1 (mod 4). - Hugo Pfoertner, May 01 2021

cp's OEIS Frontend

User: Zachary DeStefano

A370279 Indices of record high values in A370278.

Views

Author

Keywords

Examples

Crossrefs

A370278 Difference between the bound provided by Dirichlet's Simultaneous Approximation Theorem applied to Z_n (for d=3) and the best possible bound.

Views

Author

Keywords

Comments

Examples

Crossrefs

A370277 Numbers k with the property that Dirichlet's Simultaneous Approximation Theorem applied to Z_k is tight (for d = 3).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

A363069 Size of the largest subset of {1,2,...,n} such that no two elements sum to a perfect square.

Views

Author

Keywords

Examples

Links

Crossrefs

Formula

A361340 a(n) = smallest number with the property that the split-and-multiply technique (see A361338) in base n can produce all n single-digit numbers.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Python

Python

Extensions

A358177 Number of Eulerian orientations of a (labeled) 2n-dimensional hypercube graph, Q_2n. Q_2n is also the n-dimensional torus grid graph (C_4)^n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Formula

Extensions

A350547 Maximum size of a set of points taken from a hexagonal section of a hexagonal grid with side length n such that no three selected points form an equilateral triangle.

Views

Author

Keywords

Comments

Examples

Crossrefs

A343828 Numbers which are the product of two S-primes (A057948) in exactly three ways.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

Formula

A343827 Numbers which are the product of two S-primes (A057948) in exactly two ways.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs