Serguei Zolotov - cp's OEIS frontend

A341339 Square array read by descending antidiagonals where the row n (n >= 2) and column k (k >= 1) contains the largest number not greater than 2^k that has exactly n divisors, or 0 if such a number does not exist.

2, 3, 0, 7, 4, 0, 13, 4, 0, 0, 31, 9, 8, 0, 0, 61, 25, 15, 0, 0, 0, 127, 49, 27, 16, 0, 0, 0, 251, 121, 62, 16, 12, 0, 0, 0, 509, 169, 125, 16, 32, 0, 0, 0, 0, 1021, 361, 254, 81, 63, 0, 0, 0, 0, 0, 2039, 961, 511, 81, 124, 64, 30, 0, 0, 0, 0, 4093, 1849, 1018, 81, 245, 64, 56, 0, 0, 0, 0, 0

Offset: 2

Serguei Zolotov, Apr 27 2021

First row contains largest prime not greater than 2^k (where k is a column number starting with 1). Second row contains largest square of prime not greater than 2^k.

Diagonal of the square array contains sequential powers of 2 since 2^k has exactly k+1 divisors.

			Array begins:
     k = 1   2   3    4    5    6    7    8    9    10    11    12
-------------------------------------------------------------
n = 2  | 2,  3,  7,  13,  31,  61, 127, 251, 509, 1021, 2039, 4093, ...
n = 3  | 0,  4,  4,   9,  25,  49, 121, 169, 361,  961, 1849, 3721, ...
n = 4  | 0,  0,  8,  15,  27,  62, 125, 254, 511, 1018, 2047, 4087, ...
n = 5  | 0,  0,  0,  16,  16,  16,  81,  81,  81,  625,  625, 2401, ...
n = 6  | 0,  0,  0,  12,  32,  63, 124, 245, 508, 1017, 2043, 4084, ...
n = 7  | 0,  0,  0,   0,   0,  64,  64,  64,  64,  729,  729,  729, ...
n = 8  | 0,  0,  0,   0,  30,  56, 128, 255, 506, 1023, 2037, 4094, ...
n = 9  | 0,  0,  0,   0,   0,  36, 100, 256, 484,  676, 1521, 3844, ...
n = 10 | 0,  0,  0,   0,   0,  48, 112, 208, 512,  976, 2032, 4016, ...
n = 11 | 0,  0,  0,   0,   0,   0,   0,   0,   0, 1024, 1024, 1024, ...
n = 12 | 0,  0,  0,   0,   0,  60, 126, 234, 500, 1014, 2048, 4086, ...
n = 13 | 0,  0,  0,   0,   0,   0,   0,   0,   0,    0,    0, 4096, ...
...

Serguei Zolotov, Antidiagonals of table of n, a(n) for n = 2..2081
Serguei Zolotov, Python script to generate A341339 combining different methods

import sympy
# k = 1,2,3,...
# n = 2,3,4,...
def a(k, n):
    a = 2**k
    while a > 0 and sympy.divisor_count(a) != n:
        a = a - 1
    return a

A340458 Minimum length of the string over the alphabet of 3 or more symbols that has exactly n substring palindromes. Substrings are counted as distinct if they start at different offsets.

Original entry on oeis.org

1, 2, 2, 3, 4, 3, 4, 5, 5, 4, 5, 6, 6, 7, 5, 6, 7, 7, 8, 8, 6, 7, 8, 8, 9, 9, 9, 7, 8, 9, 9, 10, 10, 10, 11, 8, 9, 10, 10, 11, 11, 11, 12, 12, 9, 10, 11, 11, 12, 12, 12, 13, 13, 14, 10, 11, 12, 12, 13, 13, 13, 14, 14, 15, 14, 11, 12, 13, 13, 14, 14, 14, 15, 15, 16, 15, 16, 12, 13, 14, 14, 15, 15, 15, 16, 16, 17, 16, 17, 17

Offset: 1

Serguei Zolotov, Feb 13 2021

nonn

The uploaded Python script uses G. Manacher's algorithm to efficiently calculate the number of palindromes.

			The string AAA with length 3 has 6 palindromic substrings:
    A starting at offset 1,
    A starting at offset 2,
    A starting at offset 3,
   AA starting at offset 1,
   AA starting at offset 2,
  AAA starting at offset 1.
There is no shorter string with exactly 6 substring palindromes. So a(6) = 3.

Serguei Zolotov, Table of n, a(n) for n = 1..5000
Glenn Manacher, A new linear-time "on-line" algorithm for finding the smallest initial palindrome of a string, Journal of the ACM, (1975) 22 (3): 346-351.
Serguei Zolotov, Table of n, a(n), sample string for n = 1..5000
Serguei Zolotov, Python script to generate b-file and a-file

a(k*(k+1)/2) = k, from a string of k identical symbols.

A325159 Denominators of convergents to Pi using best rational approximation whose denominator is between consecutive powers of 2: [2^n, 2^(n+1)-1], where n = 0, 1, 2, ...

Original entry on oeis.org

1, 2, 7, 14, 21, 57, 113, 226, 339, 565, 1130, 2147, 4181, 8249, 32763, 33215, 99532, 199064, 364913, 995207, 1725033, 3450066, 5175099, 15160384, 25510582, 52746197, 131002976, 209259755, 340262731, 811528438, 1963319607, 3926639214, 6701487259, 13402974518, 20104461777

Offset: 0

Serguei Zolotov, Apr 04 2019

nonn

			The convergents are 3/1, 6/2, 22/7, 44/14, 66/21, 179/57, 355/113, 710/226, 1065/339, 1775/565, 3550/1130, 6745/2147, 13135/4181, 25915/8249, 102928/32763, ... = A325158/A325159.

Serguei Zolotov, Table of n, a(n) for n = 0..1023
E. Charrier and L. Buzer, Approximating a real number by a rational number with a limited denominator: A geometric approach, Discrete Applied Mathematics, Volume 157, Issue 16, 28 August 2009, Pages 3473-3484.
Serguei Zolotov, Python program to calculate b-file

Cf. A325158 (numerators), A002485.

A325158 Numerators of convergents to Pi using best rational approximation whose denominator is between consecutive powers of 2: [2^n, 2^(n+1)-1], where n = 0, 1, 2, ...

Original entry on oeis.org

3, 6, 22, 44, 66, 179, 355, 710, 1065, 1775, 3550, 6745, 13135, 25915, 102928, 104348, 312689, 625378, 1146408, 3126535, 5419351, 10838702, 16258053, 47627751, 80143857, 165707065, 411557987, 657408909, 1068966896, 2549491779, 6167950454, 12335900908, 21053343141, 42106686282

Offset: 0

Serguei Zolotov, Apr 04 2019

nonn

			The convergents are 3/1, 6/2, 22/7, 44/14, 66/21, 179/57, 355/113, 710/226, 1065/339, 1775/565, 3550/1130, 6745/2147, 13135/4181, 25915/8249, 102928/32763, ... = A325158/A325159.

Serguei Zolotov, Table of n, a(n) for n = 0..1023
E. Charrier and L. Buzer, Approximating a real number by a rational number with a limited denominator: A geometric approach, Discrete Applied Mathematics, Volume 157, Issue 16, 28 August 2009, Pages 3473-3484.
Serguei Zolotov, Python program to calculate b-file

Cf. A325159 (denominators), A002485.

A306727 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals: A(n,k) is the number of partitions of 3*n into powers of 3 less than or equal to 3^k.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 2, 3, 1, 1, 2, 3, 4, 1, 1, 2, 3, 5, 5, 1, 1, 2, 3, 5, 7, 6, 1, 1, 2, 3, 5, 7, 9, 7, 1, 1, 2, 3, 5, 7, 9, 12, 8, 1, 1, 2, 3, 5, 7, 9, 12, 15, 9, 1, 1, 2, 3, 5, 7, 9, 12, 15, 18, 10, 1, 1, 2, 3, 5, 7, 9, 12, 15, 18, 22, 11, 1, 1, 2, 3, 5, 7, 9, 12, 15, 18, 23, 26, 12, 1

Offset: 0

Serguei Zolotov, Mar 06 2019

Column sequences converge to A005704.

			A(3,3) = 5, because there are 5 partitions of 3*3=9 into powers of 3 less than or equal to 3^3=9: [9], [3,3,3], [3,3,1,1,1], [3,1,1,1,1,1,1], [1,1,1,1,1,1,1,1,1].
Square array A(n,k) begins:
  1, 1, 1, 1, 1, 1,  ...
  1, 2, 2, 2, 2, 2,  ...
  1, 3, 3, 3, 3, 3,  ...
  1, 4, 5, 5, 5, 5,  ...
  1, 5, 7, 7, 7, 7,  ...
  1, 6, 9, 9, 9, 9,  ...

Serguei Zolotov, Table of n, a(n) for n = 0..10584

Main diagonal gives A005704.

A181322 gives array for base p=2.

nmax = 12;
f[k_] := f[k] = 1/(1-x) 1/Product[1-x^(3^j), {j, 0, k-1}] + O[x]^(nmax+1) // CoefficientList[#, x]&;
A[n_, k_] := f[k][[n+1]];
Table[A[n-k, k], {n, 0, nmax}, {k, n, 0, -1}] // Flatten (* Jean-François Alcover, Nov 20 2019 *)

def aseq(p, x, k):
    # generic algorithm for any p - power base, p=3 for this sequence
    if x < 0:
        return 0
    if x < p:
        return 1
    # coefficients
    arr = [0]*(x+1)
    arr[0] = 1
    m = p**k
    while m > 0:
        for i in range(m, x+1, m):
            arr[i] += arr[i-m]
        m //= p
    return arr[x]
def A(n, k):
    p = 3
    return aseq(p, p*n, k)
# A(n, k), 5 = A(3, 3) = aseq(3, 3*3, 3)
# Serguei Zolotov, Mar 13 2019

G.f. of column k: 1/(1-x) * 1/Product_{j=0..k-1} (1 - x^(3^j)).

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User: Serguei Zolotov

A341339 Square array read by descending antidiagonals where the row n (n >= 2) and column k (k >= 1) contains the largest number not greater than 2^k that has exactly n divisors, or 0 if such a number does not exist.

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A340458 Minimum length of the string over the alphabet of 3 or more symbols that has exactly n substring palindromes. Substrings are counted as distinct if they start at different offsets.

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A325159 Denominators of convergents to Pi using best rational approximation whose denominator is between consecutive powers of 2: [2^n, 2^(n+1)-1], where n = 0, 1, 2, ...

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A325158 Numerators of convergents to Pi using best rational approximation whose denominator is between consecutive powers of 2: [2^n, 2^(n+1)-1], where n = 0, 1, 2, ...

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A306727 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals: A(n,k) is the number of partitions of 3*n into powers of 3 less than or equal to 3^k.

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