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.

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A236631 Triangle read by rows: T(j,k), j>=1, k>=1, in which column k lists the positive squares repeated k-1 times, except the column 1 which is A123327. The elements of the even-indexed columns are multiplied by -1. The first element of column k is in row k(k+1)/2.

Original entry on oeis.org

1, 3, 5, -1, 8, -1, 10, -4, 15, -4, 1, 16, -9, 1, 23, -9, 1, 25, -16, 4, 31, -16, 4, -1, 34, -25, 4, -1, 45, -25, 9, -1, 42, -36, 9, -1, 55, -36, 9, -4, 60, -49, 16, -4, 1, 67, -49, 16, -4, 1, 69, -64, 16, -4, 1, 86, -64, 25, -9, 1, 84, -81, 25, -9, 1, 103
Offset: 1

Views

Author

Omar E. Pol, Jan 29 2014

Keywords

Comments

T(j,k) which row j has length A003056(j) hence the first element of column k is in row A000217(j).
Row sums give A000203.
Interpreted as a sequence with index n this is also the first differences of A236630. If a(n) is positive then a(n) is the number of cells turned ON at n-th stage in the structure of A236630. If a(n) is negative then a(n) is the number of cells turned OFF at n-th stage in the structure of A236630.

Examples

			Written as an irregular triangle the sequence begins:
1;
3;
5,     -1;
8,     -1;
10,    -4;
15,    -4,    1;
16,    -9,    1;
23,    -9,    1;
25,   -16,    4;
31,   -16,    4,   -1;
34,   -25,    4,   -1;
45,   -25,    9,   -1;
42,   -36,    9,   -1;
55,   -36,    9,   -4;
60,   -49,   16,   -4,   1;
67,   -49,   16,   -4,   1;
69,   -64,   16,   -4,   1;
86,   -64,   25,   -9,   1;
84,   -81,   25,   -9,   1;
103,  -81,   25,   -9,   4;
102, -100,   36,   -9,   4,  -1;
113, -100,   36,  -16,   4,  -1;
122, -121,   36,  -16,   4,  -1;
145, -121,   49,  -16,   4,  -1;
...
For j = 15 the divisors of 15 are 1, 3, 5, 15, therefore the sum of divisors of 15 is 1 + 3 + 5 + 15 = 24. On the other hand the 15th row of triangle is 60, -49, 16, -4, 1, therefore the row sum is 60 - 49 + 16 - 4 + 1 = 24, equalling the sum of divisors of 15.

Links

Crossrefs

Cf. A000203, A000217, A000290, A003056, A004125, A024916, A008794, A123327, A196020, A211547, A236104, A236630, A237593.

Formula

T(n,1) = A000203(n) + A004125(n).

A162383 Duplicate of A123327.

Original entry on oeis.org

1, 3, 5, 8, 10, 15, 16, 23, 25, 31, 34, 45, 42, 55, 60, 67, 69, 86
Offset: 1

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Author

Keywords

Examples

			1(=1+0), 3(=3+0), 5(=4+1), 8(=7+1), 10(=6+4), 15(=12+3), 16(=8+8), etc.

Formula

a(n) = A000290(n) - A024916(n-1), n > 1. - Omar E. Pol, Jan 29 2014

A244048 Antisigma(n) minus the sum of remainders of n mod k, for k = 1,2,3,...,n.

Original entry on oeis.org

0, 0, 1, 2, 5, 6, 12, 13, 20, 24, 32, 33, 49, 50, 60, 69, 84, 85, 106, 107, 129, 140, 154, 155, 191, 197, 213, 226, 254, 255, 297, 298, 329, 344, 364, 377, 432, 433, 455, 472, 522, 523, 577, 578, 618, 651, 677, 678, 754, 762, 805, 826
Offset: 1

Views

Author

Omar E. Pol, Jun 23 2014

Keywords

Comments

For n > 1 a(n) is the sum of all aliquot parts of all positive integers < n. - Omar E. Pol, Mar 27 2021

Examples

			From _Omar E. Pol_, Mar 27 2021: (Start)
The following diagrams show a square dissection into regions that are the symmetric representation of A000203, A004125, A153485 and this sequence.
In order to construct every diagram we use the following rules:
At stage 1 in the first quadrant of the square grid we draw the symmetric representation of sigma(n) using the two Dyck paths described in the rows n and n-1 of A237593.
At stage 2 we draw a pair of orthogonal line segments (if it's necessary) such that in the drawing appears totally formed a square n X n. The area of the region that is above the symmetric representation of sigma(n) equals A004125(n).
At stage 3 we draw a zig-zag path with line segments of length 1 from (0,n-1) to (n-1,0) such that appears a staircase with n-1 steps. The area of the region (or regions) that is below the symmetric representation of sigma(n) and above the staircase equals a(n).
At stage 4 we draw a copy of the symmetric representation of A004125(n) rotated 180 degrees such that one of its vertices is the point (0,0). The area of the region (or regions) that is above of this region and below the staircase equals A153485(n).
Illustration for n = 1..6:
.                                                                    _ _ _ _ _ _
.                                                     _ _ _ _ _     |_ _ _  |_ R|
.                                        _ _ _ _ R   |_ _S_|  R|    | |_T | S |_|
.                             _ _ _ R   |_ _  |_|    | |_  |_ _|    |   |_|_ _  |
.                    _ _     |_S_|_|    | |_|_S |    |_U_|_T | |    |_  U |_T | |
.             _ S   |_ S|   U|_|_|S|    |_ U|_| |    |   | |_|S|    | |_    |_| |
.            |_|    |_|_|    |_|_|_|    |_|_ _|_|    |_V_|_U_|_|    |_V_|_ _ _|_|
.                  U        V   U       V
.
n:            1       2         3           4             5               6
R: A004125    0       0         1           1             4               3
S: A000203    1       3         4           7             6              12
T: a(n)       0       0         1           2             5               6
U: A153485    0       1         2           5             6              12
V: A004125    0       0         1           1             4               3
.
Illustration for n = 7..9:
.                                                      _ _ _ _ _ _ _ _ _
.                                _ _ _ _ _ _ _ _      |_ _ _S_ _|       |
.            _ _ _ _ _ _ _      |_ _ _ _  |     |     | |_      |_ _ R  |
.           |_ _S_ _|     |     | |_    | |_ R  |     |   |_    |_ S|   |
.           | |_    |_ R  |     |   |_  |_S |_ _|     |     |_  T |_|_ _|
.           |   |_  T |_ _|     |     |_T |_ _  |     |_ _    |_      | |
.           |_ _  |_    | |     |_ _  U |_    | |     |   |  U  |_    | |
.           |   |_U |_  |S|     |   |_    |_  | |     |   |_ _    |_  |S|
.           |  V  |   |_| |     |  V  |     |_| |     |  V    |     |_| |
.           |_ _ _|_ _ _|_|     |_ _ _|_ _ _ _|_|     |_ _ _ _|_ _ _ _|_|
.
n:                 7                    8                      9
R: A004125         8                    8                     12
S: A000203         8                   15                     12
T: a(n)           12                   13                     20
U: A153485        13                   20                     24
V: A004125         8                    8                     12
.
Illustration for n = 10..12:
.                                                         _ _ _ _ _ _ _ _ _ _ _ _
.                              _ _ _ _ _ _ _ _ _ _ _     |_ _ _ _ _ _  |         |
.     _ _ _ _ _ _ _ _ _ _     |_ _ _S_ _ _|         |    | |_        | |_ _   R  |
.    |_ _ _S_ _  |       |    | |_        |      R  |    |   |_      |     |_    |
.    | |_      | |_  R   |    |   |_      |_        |    |     |_    |_  S   |   |
.    |   |_    |_ _|_    |    |     |_      |_      |    |       |_    |_    |_ _|
.    |     |_      | |_ _|    |       |_   T  |_ _ _|    |         |_ T  |_ _ _  |
.    |       |_ T  |_ _  |    |_ _ _    |_        | |    |_ _        |_        | |
.    |_ _      |_      | |    |     |_ U  |_      | |    |   |    U    |_      | |
.    |   |_ U    |_    |S|    |       |_    |_    |S|    |   |_          |_    | |
.    |     |_      |_  | |    |         |     |_  | |    |     |_ _        |_  | |
.    |  V    |       |_| |    |  V      |       |_| |    |  V      |         |_| |
.    |_ _ _ _|_ _ _ _ _|_|    |_ _ _ _ _|_ _ _ _ _|_|    |_ _ _ _ _|_ _ _ _ _ _|_|
.
n:            10                         11                          12
R: A004125    13                         22                          17
S: A000203    18                         12                          28
T: a(n)       24                         32                          33
U: A153485    32                         33                          49
V: A004125    13                         22                          17
.
Note that in the diagrams the symmetric representation of a(n) is the same as the symmetric representation of A153485(n-1) rotated 180 degrees.
The original examples (dated Jun 24 2014) were only the diagrams for n = 11 and n = 12. (End)

Links

Crossrefs

Also zero together with A153485.
Cf. A000203, A000290, A001065, A067439, A024816, A024916, A067439, A123327, A196020, A236104, A237270, A237271, A237593, A244049, A244251, A262626, A342344.

Programs

  • Mathematica
    With[{r=Range[100]},Join[{0},Accumulate[DivisorSigma[1,r]-r]]] (* Paolo Xausa, Oct 16 2023 *)
  • Python
    from math import isqrt
def A244048(n): return (-n*(n-1)-(s:=isqrt(n-1))**2*(s+1) + sum((q:=(n-1)//k)*((k<<1)+q+1) for k in range(1,s+1)))>>1 # Chai Wah Wu, Oct 22 2023

Formula

a(n) = A024816(n) - A004125(n).
a(n) = A000217(n) - A000203(n) - A004125(n).
a(n) = A024916(n) - A000203(n) - A000217(n-1).
a(n) = A000217(n) - A123327(n).
a(n) = A153485(n-1), n >= 2.
Showing 1-3 of 3 results.

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