Charles Kusniec - cp's OEIS frontend

A377865 Smallest divisor of 2n-1 greater than or equal to sqrt(2n-1).

1, 3, 5, 7, 3, 11, 13, 5, 17, 19, 7, 23, 5, 9, 29, 31, 11, 7, 37, 13, 41, 43, 9, 47, 7, 17, 53, 11, 19, 59, 61, 9, 13, 67, 23, 71, 73, 15, 11, 79, 9, 83, 17, 29, 89, 13, 31, 19, 97, 11, 101, 103, 15, 107, 109, 37, 113, 23, 13, 17, 11, 41, 25, 127, 43, 131, 19

Offset: 1

Charles Kusniec, Nov 10 2024

Odd bisection of A033677.

Cf. A377499, A219695.

a[n_]:=Module[{k=1},While[!Divisible[2n-1,k] || kStefano Spezia, Nov 17 2024 *)

a(n) = A033677(2n-1).

a(n) = A377499(n) + A219695(n).

A377864 Largest divisor of 2n-1 less than or equal to sqrt(2n-1).

Original entry on oeis.org

1, 1, 1, 1, 3, 1, 1, 3, 1, 1, 3, 1, 5, 3, 1, 1, 3, 5, 1, 3, 1, 1, 5, 1, 7, 3, 1, 5, 3, 1, 1, 7, 5, 1, 3, 1, 1, 5, 7, 1, 9, 1, 5, 3, 1, 7, 3, 5, 1, 9, 1, 1, 7, 1, 1, 3, 1, 5, 9, 7, 11, 3, 5, 1, 3, 1, 7, 9, 1, 1, 3, 11, 5, 7, 1, 1, 9, 5, 1, 3, 7, 1, 11, 1, 13, 9

Offset: 1

Charles Kusniec, Nov 10 2024

Odd bisection of A033676.

Cf. A377499, A219695.

a[n_]:=Module[{k=2n-1},While[!Divisible[2n-1,k] || k>Sqrt[2n-1] ,k--]; k]; Array[a,86] (* Stefano Spezia, Nov 17 2024 *)

a(n) = {my(d = divisors(2*n-1)); d[ceil(#d/2)]} \\ Thomas Scheuerle, Nov 17 2024

a(n) = A033676(2n-1).

a(n) = A377499(n) - A219695(n).

A377499 a(n) is the median of the divisors of 2n-1.

Original entry on oeis.org

1, 2, 3, 4, 3, 6, 7, 4, 9, 10, 5, 12, 5, 6, 15, 16, 7, 6, 19, 8, 21, 22, 7, 24, 7, 10, 27, 8, 11, 30, 31, 8, 9, 34, 13, 36, 37, 10, 9, 40, 9, 42, 11, 16, 45, 10, 17, 12, 49, 10, 51, 52, 11, 54, 55, 20, 57, 14, 11, 12, 11, 22, 15, 64, 23, 66, 13, 12, 69, 70, 25, 12, 17, 14, 75, 76

Offset: 1

Charles Kusniec, Oct 30 2024

nonn

From Rémi Guillaume, Nov 26 2024 and Dec 05 2024: (Start)
2n-1 has only odd divisors; so the sum of any two of them is even.
a(n) and A219695(n) have opposite parity.
a(n) and n have the same parity.
a(n) = sqrt(2n-1) iff 2n-1 = (2j+1)^2 for some j >= 0, iff n is a centered square (A001844(j)); in this case, the two "median" divisors coincide with 2j+1, so their mean a(n) = 2j+1 and A219695(n) = 0.
More generally, with s a nonnegative integer:
If j >= s and n is the centered square A001844(j), then a(n-2s^2) <= 2j+1 and A219695(n-2s^2) <= 2s.
If j > (s^2)/2 and n = A001844(j), then a(n-2s^2) = 2j+1 and A219695(n-2s^2) = 2s.   (P)
Basis of the proofs: 2(n-2s^2)-1 = (2j+1)^2-(2s)^2.
If j = s and n = A001844(j), then n-2s^2 = 2j+1 and 2(n-2s^2)-1 = 4j+1.
(End)

			From _Michael De Vlieger_, Nov 01 2024: (Start)
Let u = 2*n-1, let factor d <= sqrt(u) be the largest such, and let D = u/d.
For n = 2, u = 2*2-1 = 3, d = 1, D = 3, so a(2) = (1+3)/2 = 2.
For n = 5, u = 2*5-1 = 9 is a perfect square and d = D = 3, so a(5) = (3+3)/2 = 3.
For n = 8, u = 2*8-1 = 15, d = 3, D = 5, so a(8) = (3+5)/2 = 4, etc. (End)

Cf. A219695 (associated subtrahend square base forming 2n-1), A001844 (solutions of a(n)=sqrt(2n-1)), A006254 (indices of record highs).

See also A063655, A033676, A033677, A361565, A377864, A377865.

{1}~Join~Table[u = 2*n + 1; (# + u/#)/2 &@ #[[Floor[Length[#]/2] ]] &@ Divisors[u], {n, 2, 120}] (* Michael De Vlieger, Nov 01 2024 *)

from sympy import divisors
def A377499(n): return (d:=(f:=divisors(m:=(n<<1)-1))[len(f)-1>>1])+m//d>>1 # Chai Wah Wu, Nov 07 2024

a(n) = (A033677(2n-1) + A033676(2n-1))/2.
a(n) = A063655(2n-1)/2.
a(n) = sqrt((2n-1) + A219695(n)^2).
a(n) = n iff 2n-1 is 1 or prime (n is 1 or in A006254); in this case, A219695(n) = n-1.
From Rémi Guillaume, Nov 21 2024: (Start)
a(n) = A361565(2n-1).
sqrt(2n-1) <= a(n) <= n.
a(n) = (A377865(n) + A377864(n))/2.
a(n) = A377864(n) + A219695(n).
a(n) = A377865(n) - A219695(n). (End)

New name from Rémi Guillaume, Feb 19 2025

A354776 Even numbers that are the sum of two squares; also numbers which are twice the sum of two squares.

Original entry on oeis.org

0, 2, 4, 8, 10, 16, 18, 20, 26, 32, 34, 36, 40, 50, 52, 58, 64, 68, 72, 74, 80, 82, 90, 98, 100, 104, 106, 116, 122, 128, 130, 136, 144, 146, 148, 160, 162, 164, 170, 178, 180, 194, 196, 200, 202, 208, 212, 218, 226, 232, 234, 242, 244, 250, 256, 260, 272, 274, 288, 290, 292, 296, 298, 306, 314, 320

Offset: 1

N. J. A. Sloane, Jun 26 2022, following a suggestion from Charles Kusniec

nonn

This is both the even subsequence of A001481, and twice A001481. It is an easy exercise to show that these two definitions produce the same sequence.

Cf. A001481. Essentially the same as A128106.

from itertools import count, islice
from sympy import factorint
def A354776_gen(): # generator of terms
    return filter(lambda n:(lambda m:all(d & 3 != 3 or m[d] & 1 == 0 for d in m))(factorint(n//2)),count(0,2))
A354776_list = list(islice(A354776_gen(),30)) # Chai Wah Wu, Jun 27 2022

A350576 a(n) = n/A055874(n) - A055874(n).

Original entry on oeis.org

0, -1, 2, 0, 4, -1, 6, 2, 8, 3, 10, -1, 12, 5, 14, 6, 16, 3, 18, 8, 20, 9, 22, 2, 24, 11, 26, 12, 28, 7, 30, 14, 32, 15, 34, 5, 36, 17, 38, 18, 40, 11, 42, 20, 44, 21, 46, 8, 48, 23, 50, 24, 52, 15, 54, 26, 56, 27, 58, 4, 60, 29, 62, 30, 64, 19, 66, 32, 68, 33, 70, 14, 72, 35, 74

Offset: 1

Michel Marcus, Jan 07 2022, after a suggestion from Charles Kusniec

sign

Michel Marcus, Table of n, a(n) for n = 1..10000

Cf. A055874, A350509.

Cf. A005408 (odd numbers), A056737 (another difference n/d-d).

a[n_] := Module[{k = 1}, While[Divisible[n, k], k++]; k--; n/k - k]; Array[a, 100] (* Amiram Eldar, Jan 07 2022 *)

a4(n) = my(m=1); while ((n % m) == 0, m++); m - 1; \\ A055874
a(n) = my(x=a4(n)); n/x - x;

def a(n):
    m = 2
    while n%m == 0: m += 1
    return n//(m-1) - (m-1)
print([a(n) for n in range(1, 76)]) # Michael S. Branicky, Jan 07 2022

a(n) = A350509(n) - A055874(n).

a(n) = n-1 if n is odd.

A350509 a(n) = n/A055874(n).

Original entry on oeis.org

1, 1, 3, 2, 5, 2, 7, 4, 9, 5, 11, 3, 13, 7, 15, 8, 17, 6, 19, 10, 21, 11, 23, 6, 25, 13, 27, 14, 29, 10, 31, 16, 33, 17, 35, 9, 37, 19, 39, 20, 41, 14, 43, 22, 45, 23, 47, 12, 49, 25, 51, 26, 53, 18, 55, 28, 57, 29, 59, 10, 61, 31, 63, 32, 65, 22, 67, 34, 69, 35, 71, 18, 73, 37, 75

Offset: 1

Michel Marcus, Jan 02 2022, after a suggestion from Charles Kusniec

nonn

Michel Marcus, Table of n, a(n) for n = 1..10000

Cf. A055874.

a[n_] := Module[{k = 1}, While[Divisible[n, k], k++]; k--; n/k]; Array[a, 100] (* Amiram Eldar, Jan 02 2022 *)

a(n) = my(m = 1); while ((n % m) == 0, m++); n/(m-1);

def a(n):
    m = 2
    while n%m == 0: m += 1
    return n//(m-1)
print([a(n) for n in range(1, 80)]) # Michael S. Branicky, Jan 02 2022

A350380 Triangle read by rows in which row n lists A014963(d), the exponential of Mangoldt function, for each divisor d of n.

Original entry on oeis.org

1, 1, 2, 1, 3, 1, 2, 2, 1, 5, 1, 2, 3, 1, 1, 7, 1, 2, 2, 2, 1, 3, 3, 1, 2, 5, 1, 1, 11, 1, 2, 3, 2, 1, 1, 1, 13, 1, 2, 7, 1, 1, 3, 5, 1, 1, 2, 2, 2, 2, 1, 17, 1, 2, 3, 1, 3, 1, 1, 19, 1, 2, 2, 5, 1, 1, 1, 3, 7, 1, 1, 2, 11, 1, 1, 23, 1, 2, 3, 2, 1, 2, 1, 1, 1, 5, 5, 1, 2, 13, 1

Offset: 1

Michel Marcus, Dec 28 2021, following a suggestion from Charles Kusniec

			Triangle begins:
  1;
  1, 2;
  1, 3;
  1, 2, 2;
  1, 5;
  1, 2, 3, 1;
  1, 7;
  1, 2, 2, 2;
  1, 3, 3;
  1, 2, 5, 1;
  ...

Michel Marcus, Table of n, a(n) for n = 1..10006 (rows 1 to 1358, flattened).

Cf. A014963, A027750.

Cf. A000027 (row products), A140255 (row sums).

Table[Exp[MangoldtLambda[Divisors[n]]], {n, 1, 26}] // Flatten (* Amiram Eldar, Dec 28 2021 *)

M(n) = ispower(n, , &n); if(isprime(n), n, 1); \\ A014963
row(n) = apply(M, divisors(n));

a(n) = A014963(A027750(n)).

A319135 Irregular triangle read by rows in which row n lists the positive divisors of n that are <= sqrt(n) in decreasing order.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 3, 1, 2, 1, 1, 3, 2, 1, 1, 2, 1, 3, 1, 4, 2, 1, 1, 3, 2, 1, 1, 4, 2, 1, 3, 1, 2, 1, 1, 4, 3, 2, 1, 5, 1, 2, 1, 3, 1, 4, 2, 1, 1, 5, 3, 2, 1, 1, 4, 2, 1, 3, 1, 2, 1, 5, 1, 6, 4, 3, 2, 1, 1, 2, 1, 3, 1, 5, 4, 2, 1, 1, 6, 3, 2, 1

Offset: 1

Charles Kusniec, Feb 27 2021

Reversing rows gives A161906.

			Triangle begins:
1
1
1
2 1
1
2 1
1
2 1
3 1
2 1
1
3 2 1
1
2 1
3 1
4 2 1

Cf. A038548 (row widths), A033676 (first column), A333750 (second column), A161906 (rows reversed), A027750, A056538, A061017, A340791, A340792.

row(n) = Vecrev(select(x->(x<=sqrt(n)), divisors(n))); \\ Jinyuan Wang, Mar 13 2021

a(n) * A161908(n) = A340792(n)
A161906(n) * A340791(n) = A340792(n)
A027750(n) * A056538(n) = A061017(n)

More terms from Jinyuan Wang, Mar 13 2021

A340792 List in which n appears ceiling(d(n)/2) = A038548(n) times, where d(n) is the number of divisors of n.

Original entry on oeis.org

1, 2, 3, 4, 4, 5, 6, 6, 7, 8, 8, 9, 9, 10, 10, 11, 12, 12, 12, 13, 14, 14, 15, 15, 16, 16, 16, 17, 18, 18, 18, 19, 20, 20, 20, 21, 21, 22, 22, 23, 24, 24, 24, 24, 25, 25, 26, 26, 27, 27, 28, 28, 28, 29, 30, 30, 30, 30, 31, 32, 32, 32, 33, 33, 34, 34, 35, 35, 36, 36, 36, 36, 36

Offset: 1

Charles Kusniec, Jan 21 2021

The numbers in A075362 arranged in numerical order.

			Array begins:
   1
   2
   3
   4  4
   5
   6  6
   7
   8  8
   9  9
  10 10
  11
  12 12 12
  13
  14 14
  15 15
  16 16 16
  17
  18 18 18
  19
  20 20 20
  21 21
  22 22
  23
  24 24 24 24

Charles Kusniec, Bijection between OEIS sequence A340792 and the positive Triangular Multiplication Table

Cf. A038548 (row lengths), A075362, A161906, A340791, A061017 (comparable array).

row(n) = my(d=divisors(n), r1=select(x->(x<=sqrt(n)), d), r2=Vecrev(select(x->(x>=sqrt(n)), d))); vector(#r1, k, r1[k]*r2[k]); \\ Michel Marcus, Jan 22 2021

T(n,k) = A161906(n,k) * A340791(n,k).

A340791 Irregular triangle read by rows in which row n lists the positive divisors of n that are >= sqrt(n) in decreasing order.

Original entry on oeis.org

1, 2, 3, 4, 2, 5, 6, 3, 7, 8, 4, 9, 3, 10, 5, 11, 12, 6, 4, 13, 14, 7, 15, 5, 16, 8, 4, 17, 18, 9, 6, 19, 20, 10, 5, 21, 7, 22, 11, 23, 24, 12, 8, 6, 25, 5, 26, 13, 27, 9, 28, 14, 7, 29, 30, 15, 10, 6, 31, 32, 16, 8, 33, 11, 34, 17, 35, 7, 36, 18, 12, 9, 6, 37, 38, 19

Offset: 1

Charles Kusniec, Jan 21 2021

Reversing rows gives A161908.

			Triangle begins:
   1
   2
   3
   4  2
   5
   6  3
   7
   8  4
   9  3
  10  5
  11
  12  6  4
  13
  14  7
  15  5
  16  8  4
  17
  18  9  6
  19
  20 10  5
  21  7
  22 11
  23
  24 12  8  6

Cf. A038548 (row widths), A160180 (second column), A161908 (rows reversed), A340792.

row(n) = Vecrev(select(x->(x>=sqrt(n)), divisors(n))); \\ Michel Marcus, Jan 22 2021

cp's OEIS Frontend

User: Charles Kusniec

A377865 Smallest divisor of 2n-1 greater than or equal to sqrt(2n-1).

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A377864 Largest divisor of 2n-1 less than or equal to sqrt(2n-1).

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A377499 a(n) is the median of the divisors of 2n-1.

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A354776 Even numbers that are the sum of two squares; also numbers which are twice the sum of two squares.

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A350576 a(n) = n/A055874(n) - A055874(n).

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A350509 a(n) = n/A055874(n).

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A350380 Triangle read by rows in which row n lists A014963(d), the exponential of Mangoldt function, for each divisor d of n.

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A319135 Irregular triangle read by rows in which row n lists the positive divisors of n that are <= sqrt(n) in decreasing order.

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