cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Previous Showing 11-16 of 16 results.

A095808 Number of ways to write n in the form m + (m+1) + ... + (m+k-1) + (m+k) + (m+k-1) + ... + (m+1) + m with integers m>= 1, k>=1. Or, number of divisors d of 4n-1 with 0 < (d-1)^2 < 4n.

Original entry on oeis.org

0, 0, 0, 1, 0, 0, 1, 0, 1, 1, 0, 0, 1, 1, 0, 2, 0, 0, 2, 0, 0, 1, 1, 1, 2, 0, 0, 1, 1, 1, 1, 0, 0, 3, 0, 1, 2, 0, 1, 1, 0, 0, 2, 2, 0, 1, 1, 0, 3, 0, 1, 2, 0, 1, 1, 0, 0, 3, 1, 0, 2, 1, 0, 3, 1, 0, 1, 0, 2, 2, 0, 1, 1, 1, 1, 1, 0, 0, 5, 1, 1, 1, 0, 1, 1, 1, 0, 3, 1, 0, 2, 0, 1, 3, 0, 0, 2, 1, 1, 3
Offset: 1

Views

Author

Alfred Heiligenbrunner, Jun 15 2004

Keywords

Comments

n = m + (m+1) + ... + (m+k-1) + (m+k) + (m+k-1) + ... + (m+1) + m means n = k^2 + m*(2k+1) or 4n-1 = (2k+1)*(4m+2k-1). So if 4n-1 disparts into two odd factors a*b, then k = (a-1)/2, m=(n-k^2)/(2k+1) give the solution of the origin equation. We only count solutions with k^2 < n, such that m>0. This means we are taking into account only factors a < 2n+1.
Note that a(n) = 0 if 4n-1 is prime. - Alfred Heiligenbrunner, Mar 01 2016

Examples

			a(16) = 2 because 16 = 5+6+5 and 16 = 1+2+3+4+3+2+1.
The trivial case 16=16 (k=0, m=n) is not counted. The cases m=0, e.g. 16 = 0+1+2+3+4+3+2+1+0 are not counted. The cases m<0 e.g. 16 = -4+-3+-2+-1+0+1+2+3+4+5+6+5+4+3+2+1+0+-1+-2+-3+-4 are not counted.

Links

Crossrefs

Cf. A001227, A001620, A069283, A070824, A078703.

Programs

  • Maple
    seq((numtheory[tau](4*n-1)-2)/2, n=1..100); # Ridouane Oudra, Jan 18 2025
  • Mathematica
    h1 = Table[count = 0; For[k = 1, k^2 < n, k++, If[Mod[n - k^2, 2k + 1] == 0, count++ ]]; count, {n, 100}] - or - h2 = Table[Length[Select[Divisors[4n - 1], ((# - 1)^2 < 4n) &]] - 1, {n, 100}]
a[n_] := (DivisorSigma[0, 4*n-1] - 2)/2; Array[a, 100] (* Amiram Eldar, Jan 28 2025 *)
  • PARI
    a(n) = (numdiv(4*n-1) - 2)/2; \\ Amiram Eldar, Jan 28 2025

Formula

From Ridouane Oudra, Jan 18 2025: (Start)
a(n) = (tau(4*n-1) - 2)/2.
a(n) = A070824(4*n-1)/2.
a(n) = A078703(n) - 1. (End)
Sum_{k=1..n} a(k) = (log(n) + 2*gamma - 5 + 4*log(2))*n/4 + O(n^(1/3)*log(n)), where gamma is Euler's constant (A001620). - Amiram Eldar, Jan 27 2025

A273164 Irregular array read by rows: T(n, k) giving in row n the divisors of nonprime numbers that are 3 (mod 4).

Original entry on oeis.org

1, 3, 5, 15, 1, 3, 9, 27, 1, 5, 7, 35, 1, 3, 13, 39, 1, 3, 17, 51, 1, 5, 11, 55, 1, 3, 7, 9, 21, 63, 1, 3, 5, 15, 25, 75, 1, 3, 29, 87, 1, 7, 13, 91, 1, 5, 19, 95, 1, 3, 9, 11, 33, 99, 1, 3, 37, 111, 1, 5, 23, 115, 1, 7, 17, 119, 1, 3, 41, 123, 1, 3, 5, 9, 15, 27, 45, 135, 1, 11, 13, 143, 1, 3, 7, 21, 49, 147, 1, 5, 31, 155
Offset: 1

Views

Author

Wolfdieter Lang, Jul 29 2016

Keywords

Comments

The length of row n is 2*A273165(n).
The number of divisors 1 and -1 (mod 4) in each row are identical, namely A273165(n). See the Jan 05 2004 Jovovic comment on A078703. For prime numbers 3 (mod 4) this is obvious. For the proof see a comment on A091236 with the Grosswald reference.
From Paul Curtz, Jul 31 2016: (Start)
For each row n of length 2*r(n) one has:
T(n, m)*T(n, 2*r(n)-m+1) = T(n, 2*r(n)),for m=1, 2, ... , r(n).
From the second comment it follows that the row sums are congruent to 0 modulo 4. (End)

Examples

			The irregular array T(n, k) begins:
n\k 1  2  3   4  5   6  7   8 ...
1:  1  3  5  15
2:  1  3  9  27
3:  1  5  7  35
4:  1  3 13  39
5:  1  3 17  51
6:  1  5 11  55
7:  1  3  7   9 21  63
8:  1  3  5  15 25  75
9:  1  3 29  87
10: 1  7 13  91
11: 1  5 19  95
12: 1  3  9  11 33  99
13: 1  3 37 111
14: 1  5 23 115
15: 1  7 17 119
16: 1  3 41 123
17: 1  3  5   9 15  27 45 135
18: 1 11 13 143
19: 1  3  7  21 49 147
20: 1 5 31 155
...
The irregular array modulo 4 gives (-1 for 3 (mod 4)):
n\k 1  2  3   4  5   6  7   8 ...
1:  1  -1  1  -1
2:  1  -1  1  -1
3:  1   1 -1  -1
4:  1  -1  1  -1
5:  1  -1  1  -1
6:  1   1 -1  -1
7:  1  -1  1  -1  1  -1
8:  1  -1  1  -1  1  -1
9:  1  -1  1  -1
10: 1  -1  1  -1
11: 1   1 -1  -1
12: 1  -1  1  -1  1  -1
13: 1  -1  1  -1
14: 1   1 -1  -1
15: 1  -1  1  -1
16: 1  -1  1  -1
17: 1  -1  1   1  -1  -1  1 -1
18: 1  -1  1  -1
19: 1  -1 -1   1   1  -1
20: 1   1 -1  -1
...

Crossrefs

Cf. A004767, A078703, A091236, A273165.

Programs

  • Mathematica
    Divisors@ Select[Range@ 155, CompositeQ@ # && Mod[#, 4] == 3 &] // Flatten (* Michael De Vlieger, Aug 01 2016 *)

Formula

T(n, k) gives the k-th divisor of A091236(n) in increasing order.

A273165 One half the number of divisors of nonprime numbers that are 3 (mod 4).

Original entry on oeis.org

2, 2, 2, 2, 2, 2, 3, 3, 2, 2, 2, 3, 2, 2, 2, 2, 4, 2, 3, 2, 2, 3, 3, 2, 2, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4
Offset: 1

Views

Author

Wolfdieter Lang, Jul 29 2016

Keywords

Comments

Row length numbers of A273164.
With the Jan 05 2004 Jovovic comment on A078703 a(n) is also the number of divisors +1 as well as -1 (mod 4) of A091236(n). See the example section of A273164.

Crossrefs

Cf. A078703, A091236, A273164.

Formula

a(n) = number of divisors of A091236(n)(nonprime numbers 3 (mod 4)).

A359289 Number of divisors of 4*n-2 of form 4*k+1.

Original entry on oeis.org

1, 1, 2, 1, 2, 1, 2, 2, 2, 1, 2, 1, 3, 2, 2, 1, 2, 2, 2, 2, 2, 1, 4, 1, 2, 2, 2, 2, 2, 1, 2, 3, 4, 1, 2, 1, 2, 3, 2, 1, 3, 1, 4, 2, 2, 2, 2, 2, 2, 3, 2, 1, 4, 1, 2, 2, 2, 2, 4, 2, 2, 2, 4, 1, 2, 1, 2, 4, 2, 1, 2, 2, 4, 3, 2, 1, 4, 2, 2, 2, 2, 1, 4, 1, 3, 3, 2, 3, 2, 1
Offset: 1

Views

Author

Seiichi Manyama, Dec 24 2022

Keywords

Crossrefs

Cf. A001826, A078703, A359227.

Programs

  • Mathematica
    a[n_] := DivisorSum[4*n-2, 1 &, Mod[#, 4] == 1 &]; Array[a, 100] (* Amiram Eldar, Aug 16 2023 *)
  • PARI
    a(n) = sumdiv(4*n-2, d, d%4==1);
  • PARI
    my(N=100, x='x+O('x^N)); Vec(sum(k=1, N, x^k/(1-x^(4*k-2))))
  • PARI
    my(N=100, x='x+O('x^N)); Vec(sum(k=1, N, x^(2*k-1)/(1-x^(4*k-3))))

Formula

a(n) = A001826(4*n-2).
G.f.: Sum_{k>0} x^k/(1 - x^(4*k-2)).
G.f.: Sum_{k>0} x^(2*k-1)/(1 - x^(4*k-3)).

A363291 Sum of divisors of 4*n-1 of form 4*k+1.

Original entry on oeis.org

1, 1, 1, 6, 1, 1, 10, 1, 6, 14, 1, 1, 18, 6, 1, 31, 1, 1, 31, 1, 1, 30, 14, 6, 43, 1, 1, 38, 6, 18, 42, 1, 1, 60, 1, 14, 71, 1, 6, 54, 1, 1, 67, 31, 1, 62, 18, 1, 84, 1, 30, 79, 1, 6, 74, 1, 1, 132, 6, 1, 91, 14, 1, 108, 38, 1, 90, 1, 31, 103, 1, 42, 98, 6, 14, 102, 1, 1, 186, 30, 18, 110, 1, 6, 114, 50, 1
Offset: 1

Views

Author

Seiichi Manyama, Jul 08 2023

Keywords

Crossrefs

Cf. A050449, A078703, A364082.

Programs

  • Mathematica
    a[n_] := DivisorSum[4*n - 1, # &, Mod[#, 4] == 1 &]; Array[a, 100] (* Amiram Eldar, Jul 08 2023 *)
  • PARI
    a(n) = sumdiv(4*n-1, d, (d%4==1)*d);

Formula

a(n) = A050449(4*n-1).
G.f.: Sum_{k>0} (4*k-3) * x^(3*k-2) / (1 - x^(4*k-3)).

A363359 Sum of divisors of 4*n-1 of form 4*k+3.

Original entry on oeis.org

3, 7, 11, 18, 19, 23, 30, 31, 42, 42, 43, 47, 54, 66, 59, 73, 67, 71, 93, 79, 83, 90, 98, 114, 113, 103, 107, 114, 138, 126, 126, 127, 131, 180, 139, 154, 157, 151, 186, 162, 163, 167, 193, 217, 179, 186, 198, 191, 252, 199, 210, 233, 211, 258, 222, 223, 227, 252, 282, 239, 273, 266, 251, 324, 266, 263, 270
Offset: 1

Views

Author

Seiichi Manyama, Jul 08 2023

Keywords

Crossrefs

Cf. A050452, A078703, A364085.

Programs

  • Mathematica
    a[n_] := DivisorSum[4*n - 1, # &, Mod[#, 4] == 3 &]; Array[a, 100] (* Amiram Eldar, Jul 08 2023 *)
  • PARI
    a(n) = sumdiv(4*n-1, d, (d%4==3)*d);

Formula

a(n) = A050452(4*n-1).
G.f.: Sum_{k>0} (4*k-1) * x^k / (1 - x^(4*k-1)).
Previous Showing 11-16 of 16 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.