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.

Showing 1-3 of 3 results.

A348389 Irregular triangle read by rows: row n gives for n >= 2 a concatenation of the finite sequences of the multiples of k, larger than k and not exceeding n, for k = 1, 2, ..., floor(n/2).

Original entry on oeis.org

2, 2, 3, 2, 3, 4, 4, 2, 3, 4, 5, 4, 2, 3, 4, 5, 6, 4, 6, 6, 2, 3, 4, 5, 6, 7, 4, 6, 6, 2, 3, 4, 5, 6, 7, 8, 4, 6, 8, 6, 8, 2, 3, 4, 5, 6, 7, 8, 9, 4, 6, 8, 6, 9, 8, 2, 3, 4, 5, 6, 7, 8, 9, 10, 4, 6, 8, 10, 6, 9, 8, 10, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 4, 6, 8, 10, 6, 9, 8, 10, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 4, 6, 8, 10, 12, 6, 9, 12, 8, 12, 10, 12
Offset: 2

Views

Author

Wolfdieter Lang, Oct 31 2021

Keywords

Comments

The length of row n is A002541(n).
The sum of row n is A348392(n).
The lengths of the sublists for these multiples of k in row n are given by T(n, k) = A348388(n, k), for n >= 2 and k = 1, 2, ..., floor(n/2).

Examples

			The irregular triangle a(n, m) begins: (the k-sublists are separated by a vertical bar)
n\m   1 2 3 4 5 6 7 8  9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 ...
-------------------------------------------------------------------------
2:    2
3:    2 3
4:    2 3 4|4
5:    2 3 4 5|4
6:    2 3 4 5 6|4 6|6
7:    2 3 4 5 6 7|4 6| 6
8:    2 3 4 5 6 7 8|4  6  8| 6| 8
9:    2 3 4 5 6 7 8 9| 4  6  8| 6  9| 8
10:   2 3 4 5 6 7 8 9 10| 4  6  8 10| 6  9| 8|10
11:   2 3 4 5 6 7 8 9 10 11| 4  6  8 10| 6  9| 8|10
12:   2 3 4 5 6 7 8 9 10 11 12| 4  6  8 10 12| 6  9 12| 8 12|10|12
13:   2 3 4 5 6 7 8 9 10 11 12 13| 4  6  8 10 12| 6  9 12| 8 12|10|12
...

Crossrefs

Cf. A002541, A348388, A348392.

Programs

  • Mathematica
    nrows=10;Table[Flatten[Table[Range[2k,n,k],{k,Floor[n/2]}]],{n,2,nrows+1}] (* Paolo Xausa, Nov 23 2021 *)

Formula

The entries a(n, m) of row n, for n > = 2 and m = 1, 2, ..., A002541(n), are given by the concatenation of the sequences k*(2, 3, ..., t(n,k)) for k = 1, 2, ..., floor(n/2), with t(n, k) = floor((n-k)/k) + 1.

A348390 Irregular triangle read by rows: for n >= 2 the row members a(n, m) give the proper divisors of k, followed by the multiples of k larger than k and not exceeding n, for k = 1, 2, ..., n.

Original entry on oeis.org

2, 1, 2, 3, 1, 1, 2, 3, 4, 1, 4, 1, 1, 2, 2, 3, 4, 5, 1, 4, 1, 1, 2, 1, 2, 3, 4, 5, 6, 1, 4, 6, 1, 6, 1, 2, 1, 1, 2, 3, 2, 3, 4, 5, 6, 7, 1, 4, 6, 1, 6, 1, 2, 1, 1, 2, 3, 1, 2, 3, 4, 5, 6, 7, 8, 1, 4, 6, 8, 1, 6, 1, 2, 8, 1, 1, 2, 3, 1, 1, 2, 4
Offset: 2

Views

Author

Wolfdieter Lang, Nov 07 2021

Keywords

Comments

The length of row n is 2*A002541(n), for n >= 2.
The sum of row n is A348391(n). The sum of the proper divisors of row n is A153485(n). The sum of the multiples in row n is A348392(n). Hence, A348391(n) = A153485(n) + A348392(n).
For k = 1 the proper divisor set is empty, and for k > floor(n/2) the set of multiples is empty.

Examples

			The irregular triangle a(n, m), m = 1, 2, ..., 2*A002541(n) begins:
(members for k = 1, 2, ..., n are separated by a vertical bar, and the proper divisors and multiples are separated by a comma)
n\m 1 2 3 4 5 6 7 8  9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 ...
-----------------------------------------------------------------------------------
2:  2|1
3:  2 3|1|1
4:  2 3 4|1,4|1|1 2
5:  2 3 4 5|1,4|1|1  2| 1
6:  2 3 4 5 6|1,4 6| 1, 6| 1  2| 1| 1 2 3
7:  2 3 4 5 6 7|1,4  6| 1, 6| 1  2| 1| 1  2  3| 1
8:  3 4 5 6 7 8|1,4  6  8| 1 ,6| 1  2 ,8| 1| 1  2  3| 1| 1  2  4
9:  2 3 4 5 6 7 8 9| 1, 4  6  8| 1, 6  9| 1  2, 8| 1| 1  2  3| 1| 1  2  4| 1  3
...
n = 10: 2 3 4 5 6 7 8 9 10 | 1, 4 6 8 10 | 1, 6 9 | 1 2, 8 | 1, 10 | 1 2 3 | 1 | 1 2 4 | 1 3 | 1 2 5
-----------------------------------------------------------------------------------
n = 4:  d(4, 1) = {}, m(4, 1) = {2, 3, 4}; d(4, 2) = {1}, m(4, 2) = {4}; d(4, 3) = {1}, m(4, 3) = {}; d(4, 4) = {1, 2}, m(4, 4) = {}, This explains row n = 4.

Crossrefs

Cf. A002541, A027751, A153485, A348389, A348391, A348392.

Programs

  • Mathematica
    nrows=10;Table[Flatten[Table[Join[Most[Divisors[k]],Range[2k,n,k]],{k,n}]],{n,2,nrows+1}] (* Paolo Xausa, Nov 23 2021 *)

Formula

For n >= 2 row n gives the sequence of the sequence d(n, k) of proper divisors of k (A027751(k)) followed by the sequences m(n, k) of the multiples of k, larger than k and not exceeding n (A348389), for k = 1, 2, 3, ..., n.

A348391 Row sums of irregular triangle A348390.

Original entry on oeis.org

3, 7, 18, 24, 48, 56, 87, 109, 147, 159, 235, 249, 301, 355, 434, 452, 563, 583, 705, 779, 859, 883, 1087, 1143, 1237, 1331, 1499, 1529, 1781, 1813, 2004, 2118, 2240, 2358, 2701, 2739, 2875, 3009, 3339, 3381, 3729, 3773
Offset: 2

Views

Author

Wolfdieter Lang, Nov 07 2021

Keywords

Examples

			a(5) = 2 + 3 + 4 + 5 + 1 + 4 + 1 + 1 + 2 + 1 = (1 + 1 + 1 + 2 + 1) + (2 + 3 + 4 + 5 + 4) = 6 + 18 = 24.

Crossrefs

Cf. A002541, A348390.

Programs

  • Mathematica
    nterms=50;Table[Total[Flatten[Table[Join[Most[Divisors[k]],Range[2k,n,k]],{k,n}]]],{n,2,nterms+1}] (* Paolo Xausa, Nov 23 2021 *)

Formula

a(n) = Sum_{m=1..2*A002541(n)} A348390(n, m), for n >= 2.
a(n) = A153485(n) + A348392(n).
Showing 1-3 of 3 results.

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