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 21-26 of 26 results.

A361903 Numbers k for which sqrt(k/2) divides k and the symmetric representation of sigma(k) has a single part.

Original entry on oeis.org

2, 8, 18, 32, 72, 128, 162, 200, 288, 392, 450, 512, 648, 800, 882, 968, 1152, 1352, 1458, 1568, 1800, 2048, 2178, 2592, 3042, 3200, 3528, 3872, 4050, 4608, 5000, 5202, 5408, 5832, 6272, 6498, 7200, 7938, 8192, 8712, 9248, 9522, 9800, 10368, 11250, 11552, 12168, 12800, 13122, 14112
Offset: 1

Views

Author

Hartmut F. W. Hoft, Mar 28 2023

Keywords

Comments

Every number in this sequence has the form 2^(2*i + 1) * k^(2*j), i,j>=0, k>=1.
The number of 1's in row a(n) of the triangle in A237048 as well as the length of that row are odd.

Examples

			a(4) = 32 has 4 as its single middle divisor, and its symmetric representation of sigma consists of one part of width 1.
a(9) = 288 = 2^5 * 3^2 has 3 middle divisors - 12 = 2^2 * 3 , 16 = 2^4, 18 = 2 * 3^2 - and its symmetric representation of sigma consists of one part, the section of maximum width 3 of the single part includes the diagonal (see also A250068).

Crossrefs

Intersection of A001105 and A174973.
Subsequence of A071562 and of A319796.
Cf. A235791, A237048, A237270, A237271, A237593, A250068.

Programs

  • Mathematica
    (* Function a237271[ ] is defined in A237271 *)
a361903[n_] := Select[Range[n], IntegerQ[#/Sqrt[#/2]]&&a237271[#]==1&]
a361903[15000]

Formula

a(n) = k when A001105(n) = k and A237271(k) = 1.

A361905 Numbers k for which sqrt(k/2) divides k and the width at the diagonal of the symmetric representation of sigma(k) equals 1.

Original entry on oeis.org

2, 8, 18, 32, 50, 98, 128, 162, 200, 242, 338, 392, 512, 578, 722, 882, 968, 1058, 1250, 1352, 1458, 1682, 1922, 2048, 2178, 2312, 2738, 2888, 3042, 3362, 3698, 3872, 4232, 4418, 4802, 5000, 5202, 5408, 5618, 6050, 6498, 6728, 6962, 7442, 7688, 8192, 8450, 8978, 9248, 9522
Offset: 1

Views

Author

Hartmut F. W. Hoft, Mar 28 2023

Keywords

Comments

Every number in this sequence has the form 2^(2*i + 1) * k^(2*j), i, j >= 0, k >= 1.
The number of 1's in row a(n) of the triangle in A237048 as well as the length of that row are odd.

Examples

			a(4) = 32 has 4 as its single middle divisor, and its symmetric representation of sigma consists of one part of width 1.
a(5) = 50 has 5 as its single middle divisor, and its symmetric representation of sigma consists of three parts of width 1.
a(9) = 200 = 2^3 * 5^2 has 10 = 2 * 5 as its single middle divisor, and its symmetric representation of sigma consists of one part of maximum width 2 (A250068), but has width 1 at the diagonal.
a(39) = 6050 = 2^1 * 5^2 * 11^2 has 55 as its single middle divisor; it is the first number in the sequence whose symmetric representation of sigma consists of 3 parts and its central part has maximum width 2, but has width 1 at the diagonal.

Crossrefs

Intersection of A001105 and A320137.
Subsequence of A071562 and of A319796.
Cf. A004171, A235791, A237048, A237270, A237271, A237593, A250068.

Programs

  • Mathematica
    (* Function a249223[ ] is defined in A320137 *)
a361905[n_] := Select[Range[n], IntegerQ[#/Sqrt[#/2]]&&Last[a249223[#]]==1&]
a361905[10000]

Formula

a(n) = k when A001105(n) = k and A320137(k) = 1.

A375611 Numbers k whose symmetric representation of sigma(k) has at least a part with maximum width 2.

Original entry on oeis.org

6, 12, 15, 18, 20, 24, 28, 30, 35, 36, 40, 42, 45, 48, 54, 56, 63, 66, 70, 75, 77, 78, 80, 88, 91, 96, 99, 100, 102, 104, 105, 108, 110, 112, 114, 117, 130, 132, 135, 138, 143, 150, 153, 154, 156, 160, 162, 165, 170, 174, 175, 176, 182, 186, 187, 189, 190, 192, 195, 196, 200
Offset: 1

Views

Author

Hartmut F. W. Hoft, Aug 21 2024

Keywords

Comments

Number m = 2^k * q, k >= 0 and q odd, is in this sequence precisely when for any divisor s <= A003056(m) of q there is at most one divisor t of q satisfying s < t <= min(2^(k+1) * s, A003056(m)), and at least one such pair s < t of successive odd divisors exists. Equivalently, row m of the triangle in A249223 contains at least one 2, but no number larger than 2.

Examples

			a(4) = 18 has width pattern 1 2 1 2 1 in its symmetric representation of sigma consisting of a single part, and row 18 in the triangle of A249223 is 1 1 2 1 1.
a(9) = 35 has width pattern 1 0 1 2 1 0 1 in its symmetric representation of sigma consisting of 3 parts, and row 35 in the triangle of A249223 is 1 0 0 0 1 1 2.
Irregular triangle of rows a(n) in triangle of A341970, i.e. of positions of 1's in triangle of A237048, and for the corresponding widths to the diagonal in triangle of A341969:
a(n)| row in A341970      left half of row in A341969
6   | 1   3               1   2
12  | 1   3               1   2
15  | 1   2   3   5       1   0   1   2
18  | 1   3   4           1   2   1
20  | 1   5               1   2
24  | 1   3               1   2
28  | 1   7               1   2
30  | 1   3   4   5       1   2   1   2
35  | 1   2   5   7       1   0   1   2
36  | 1   3   8           1   2   1
...

Crossrefs

Column 2 of A253258.
Subsequence of A005279.
Some subsequences are A352030, A370205, A370206, A370209.
Cf. A003056, A174905, A235791, A237593, A249223, A249351, A250068, A341969, A341970.

Programs

  • Mathematica
    eP[n_] := If[EvenQ[n], FactorInteger[n][[1, 2]], 0]+1
sDiv[n_] := Module[{d=Select[Divisors[n], OddQ]}, Select[Union[d, d*2^eP[n]], #<=row[n]&]]
mW2Q[n_] := Max[FoldWhileList[#1+If[OddQ[#2], 1, -1]&, sDiv[n], #1<=2&]]==2
a375611[m_, n_] := Select[Range[m, n], mW2Q]
a375611[1, 200]

A347528 Total number of layers of width 1 of all symmetric representations of sigma() with subparts of all positive integers <= n.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 8, 9, 10, 11, 12, 14, 15, 16, 18, 19, 20, 22, 23, 25, 26, 27, 28, 30, 31, 32, 33, 35, 36, 38, 39, 40, 41, 42, 44, 46, 47, 48, 49, 51, 52, 54, 55, 56, 58, 59, 60, 62, 63, 64, 65, 66, 67, 69, 70, 72, 73, 74, 75, 78, 79, 80, 82, 83, 84, 86, 87, 88
Offset: 1

Views

Author

Omar E. Pol, Sep 05 2021

Keywords

Examples

			For the first five positive integers every symmetric representation of sigma() with subparts has only one layer of width 1, so a(5) = 1 + 1 + 1 + 1 + 1 = 5.
For n = 6 the symmetric representation of sigma(6) with subparts has two layers of width 1 as shown below:
                     _ _ _ _
                    |_ _ _  |_
                          | |_|_
                          |_ _  |
                              | |
                              | |
                              |_|
So a(6) = 5 + 2 = 7.

Links

Crossrefs

Partial sums of A250068.
Cf. A196020, A235791, A236104, A237270, A237271, A237591, A237590, A237593, A279387, A279391, A279667, A280850, A280851, A296508.

Programs

  • Mathematica
    Accumulate@ Map[Max@ Accumulate[#] &, Table[If[OddQ[k], Boole@ Divisible[n, k], -Boole@ Divisible[n - k/2, k]], {n, 68}, {k, Floor[(Sqrt[8 n + 1] - 1)/2]}]] (* Michael De Vlieger, Oct 27 2021 *)

A360022 Triangle read by rows: T(n,k) is the sum of the widths of the k-th diagonals of the symmetric representation of sigma(n).

Original entry on oeis.org

1, 1, 2, 0, 2, 2, 1, 2, 2, 2, 0, 0, 2, 2, 2, 2, 2, 2, 2, 2, 2, 0, 0, 0, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 2, 1, 2, 0, 0, 2, 2, 2, 2, 2, 0, 2, 2, 2, 2, 2, 2, 2, 2, 2, 0, 0, 0, 0, 0, 2, 2, 2, 2, 2, 2, 2, 4, 4, 2, 2, 2, 2, 2, 2, 2, 2, 2, 0, 0, 0, 0, 0, 0, 2, 2, 2, 2, 2, 2, 2, 0, 0, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2
Offset: 1

Views

Author

Omar E. Pol, Jan 22 2023

Keywords

Comments

The main diagonal of the diagram called "symmetric representation of sigma(n)" is its axis of symmetry. In this case it is also the first diagonal of the diagram. The second diagonals are the two diagonals that are adjacent to the main diagonal. The third diagonals are the two diagonals that are adjacent to the second diagonals. And so on.
If and only if n is a power of 2 (A000079) then row n lists the first n terms of A040000 (the same sequence as the right border of the triangle).
If and only if n is an odd prime (A065091) then row n lists (n - 1)/2 zeros together with 1 + (n - 1)/2 2's.
If and only if n is an even perfect number (Cf. A000396) then row n lists n 2's (the first n terms of A007395).
For further information about the mentioned "widths" see A249351.

Examples

			Triangle begins (rows: 1..16):
  1;
  1, 2;
  0, 2, 2;
  1, 2, 2, 2;
  0, 0, 2, 2, 2;
  2, 2, 2, 2, 2, 2;
  0, 0, 0, 2, 2, 2, 2;
  1, 2, 2, 2, 2, 2, 2, 2;
  1, 2, 0, 0, 2, 2, 2, 2, 2;
  0, 2, 2, 2, 2, 2, 2, 2, 2, 2;
  0, 0, 0, 0, 0, 2, 2, 2, 2, 2, 2;
  2, 4, 4, 2, 2, 2, 2, 2, 2, 2, 2, 2;
  0, 0, 0, 0, 0, 0, 2, 2, 2, 2, 2, 2, 2;
  0, 0, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2;
  2, 2, 2, 2, 0, 0, 0, 2, 2, 2, 2, 2, 2, 2, 2;
  1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2;
  ...

Links

Crossrefs

Row sums give A000203.
Column 1 gives A067742.
Right border gives A040000.
Cf. A000079, A000396, A007395, A065091, A196020, A235791, A236104, A237270, A237271, A237591, A237593, A245092, A249351, A250068, A250070, A262626.

Formula

T(n,1) = A067742(n) = A249351(n,n).
T(n,k) = 2*A249351(n,n+k-1), if 1 < k <= n.

A365265 Numbers k for which sqrt(k/2) divides k and the symmetric representation of sigma(k) consists of a single part and its width at the diagonal equals 1.

Original entry on oeis.org

2, 8, 18, 32, 128, 162, 200, 392, 512, 882, 968, 1352, 1458, 2048, 2178, 3042, 3872, 5000, 5202, 5408, 6498, 8192, 9248, 9522, 11552, 13122, 15138, 16928, 17298, 19208, 26912, 30752, 32768, 36992, 43218, 43808, 46208, 53792, 58482, 59168, 67712, 70688
Offset: 1

Views

Author

Hartmut F. W. Hoft, Aug 29 2023

Keywords

Comments

Every number a(n) has the form 2^(2*i + 1) * s^2, i>= 0 and s odd, the single middle divisor of a(n) is sqrt(a(n)/2), and sqrt(2*a(n)) - 1 = floor((sqrt(8*n + 1) - 1)/2) = A003056(a(n)).
The least number in the sequence with 3 odd prime divisors is a(126) = 1630818 = 2^1 * 3^2 * 7^2 * 43^2.
Conjecture: Let a(n) = 2^(2i+1) * s^2, i>=0 and s odd, be a number in the sequence.
(1) For any odd prime divisor p of s, number a(n) * p^2 is in the sequence.
(2) For any odd prime p not a divisor of s, number a(n) * p^2 is in the sequence if p satisfies sqrt(2*a(n)) < p < 2*a(n).

Examples

			a(5) = 128 = 2^7  has 2^3 as its single middle divisor, and its symmetric representation of sigma consists of one part of width 1.
a(10) = 882 = 2 * 3^2 * 7^2 has 3 * 7 as its single middle divisor, its symmetric representation of sigma is the smallest in this sequence of maximum width 3, consists of one part, and has width 1 at the diagonal.
A table of ranges for the single odd prime factor p for numbers k in the sequence having the form 2^(2i+1) * p^(2j), i>=0 and j>0, indexed by exponent 2i+1 of 2 in number k. The lower bound is A014210(i+1) and the upper bound is A014234(2(i+1)) = A104089(i+1):
---------------------
  2i+1  /---- p ----/
---------------------
  1       3  ..    3
  3       5  ..   13
  5      11  ..   61
  7      17  ..  251
  9      37  .. 1021
...

Crossrefs

Intersection of A361903 and A361905.
Also subsequence of the following sequences: A001105, A071562, A238443 = A174973, A319796, A320137.
The powers of 2 with an odd index (A004171) form a subsequence.
Cf. A003056, A014210, A014234, A104089, A235791, A237048, A237270, A237271, A237593, A249223, A250068.

Programs

  • Mathematica
    (* a2[ ] and its support functions are defined in A249223 *)
a365265Q[n_] := Module[{list=If[Divisible[n, Sqrt[n/2]], a2[n], {0}]}, Last[list]==1&&AllTrue[list, #>0&]]
a365265[{m_, n_}] := Select[Range[m, n], a365265Q]
a365265[{1,75000}]
Previous Showing 21-26 of 26 results.

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