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 1-5 of 5 results.

A271580 Magic sums of 4 X 4 magic squares composed of squares.

Original entry on oeis.org

2823, 3230, 4350, 4590, 6462, 7150, 7735, 7905, 8515, 9230, 9775, 9945, 10625, 11220, 11292, 11310, 11985, 12155, 12675, 12750, 12920, 13050, 13390, 13775, 14355, 14705, 14825, 15275, 15300, 15950, 16150, 16250, 16830, 17300, 17316, 17400, 18315, 18360
Offset: 1

Views

Author

Arkadiusz Wesolowski, Apr 10 2016

Keywords

Comments

The magic sum of Allan W. Johnson, Jr.'s 4 X 4 magic square of squares (A359383) is 93025. - Robert C. Lyons, Dec 30 2022
The magic sum of Euler's magic square (A358445) is 8515. - Michel Marcus, Dec 30 2022

Examples

			Square with magic constant a(1) = 2823:
  |----|----|----|----|
  | 3^2|  1 |38^2|37^2|
  |----|----|----|----|
  | 2^2|47^2|13^2|21^2|
  |----|----|----|----|
  |43^2|18^2|11^2|23^2|
  |----|----|----|----|
  |31^2|17^2|33^2|22^2|
  |----|----|----|----|
Square with magic constant a(2) = 3230:
  |----|----|----|----|
  |13^2|  1 |42^2|36^2|
  |----|----|----|----|
  | 6^2|48^2|19^2|23^2|
  |----|----|----|----|
  |44^2|22^2| 9^2|27^2|
  |----|----|----|----|
  |33^2|21^2|32^2|26^2|
  |----|----|----|----|

Links

Crossrefs

Cf. A000290, A271579, A271711, A359383, A358445. Subsequence of A271581. Supersequence of A271582.

A271711 Irregular table read by rows: n-th row lists the 16 n-gonal numbers of a 4 X 4 magic square with the smallest magic sum, and the sum of the first two terms of each row is as small as it is possible.

Original entry on oeis.org

66, 1, 300, 1035, 465, 630, 171, 136, 780, 105, 496, 21, 91, 666, 435, 210, 9, 1, 1444, 1369, 4, 2209, 169, 441, 1849, 324, 121, 529, 961, 289, 1089, 484, 115787, 24130, 1908012, 543305, 1214550, 1040417, 20475, 315792, 760060, 1526617, 3725, 300832, 500837, 70, 659022, 1431305
Offset: 3

Views

Author

Arkadiusz Wesolowski, Apr 12 2016

Keywords

Comments

The magic constants are, respectively: 1402 = A271579(1), 2823 = A271580(1), and 2591234.

Examples

			The magic squares are:
triangular numbers
|----|----|----|----|
| 66 |  1 | 300|1035|
|----|----|----|----|
| 465| 630| 171| 136|
|----|----|----|----|
| 780| 105| 496| 21 |
|----|----|----|----|
| 91 | 666| 435| 210|
|----|----|----|----|
.
square numbers
|----|----|----|----|
|  9 |  1 |1444|1369|
|----|----|----|----|
|  4 |2209| 169| 441|
|----|----|----|----|
|1849| 324| 121| 529|
|----|----|----|----|
| 961| 289|1089| 484|
|----|----|----|----|
.
pentagonal numbers
|-------|-------|-------|-------|
| 115787| 24130 |1908012| 543305|
|-------|-------|-------|-------|
|1214550|1040417| 20475 | 315792|
|-------|-------|-------|-------|
| 760060|1526617|  3725 | 300832|
|-------|-------|-------|-------|
| 500837|   70  | 659022|1431305|
|-------|-------|-------|-------|

Links

Crossrefs

Cf. A271579, A271580, A271712.

A271581 Magic sums of 4 X 4 magic squares composed of triangular numbers and squares.

Original entry on oeis.org

251, 305, 317, 341, 362, 366, 371, 377, 387, 389, 410, 421, 427, 431, 434, 435, 440, 446, 455, 458, 470, 482, 483, 484, 490, 494, 497, 500, 506, 512, 515, 519, 521, 523, 525, 536, 546, 547, 548, 550, 551, 553, 563, 566, 572, 575, 578, 580, 584, 586, 590, 591
Offset: 1

Views

Author

Arkadiusz Wesolowski, Apr 10 2016

Keywords

Examples

			Examples of 4 X 4 magic squares composed of triangular numbers and squares.
.
|---|---|---|---|
| 6 | 10|210| 25|
|---|---|---|---|
| 81|121| 21| 28|
|---|---|---|---|
| 64|105| 4 | 78|
|---|---|---|---|
|100| 15| 16|120|
|---|---|---|---|
The magic constant is 251 = a(1).
.
|---|---|---|---|
| 4 | 10| 81|210|
|---|---|---|---|
|105|144| 55| 1 |
|---|---|---|---|
|171| 15| 91| 28|
|---|---|---|---|
| 25|136| 78| 66|
|---|---|---|---|
The magic constant is 305 = a(2).

Links

Crossrefs

Cf. A271578, A271579.

A271582 Magic sums of 4 X 4 magic squares composed of odd squares.

Original entry on oeis.org

11220, 15300, 17300, 17316, 19380, 20900, 26900, 27300, 28100, 29900, 31620, 33660, 34060, 34580, 35700, 39100, 39780, 42500, 42900, 43540, 43732, 43860, 44748, 45900, 46900, 47580, 47892, 47940, 49300, 50700, 51508, 52260, 53300, 53980, 54340, 55100, 55796
Offset: 1

Views

Author

Arkadiusz Wesolowski, Apr 10 2016

Keywords

Comments

Every number of the form 8*x + 4, where x is a term of A271579, is in the sequence.

Examples

			Examples of 4 X 4 magic squares composed of odd squares.
.
|-----|-----|-----|-----|
| 23^2| 3^2 | 49^2| 91^2|
|-----|-----|-----|-----|
| 61^2| 71^2| 37^2| 33^2|
|-----|-----|-----|-----|
| 79^2| 29^2| 63^2| 13^2|
|-----|-----|-----|-----|
| 27^2| 73^2| 59^2| 41^2|
|-----|-----|-----|-----|
The magic constant is 11220 = a(1).
.
|-----|-----|-----|-----|
| 3^2 | 19^2| 59^2|107^2|
|-----|-----|-----|-----|
| 49^2|103^2| 47^2| 9^2 |
|-----|-----|-----|-----|
|113^2| 39^2| 31^2| 7^2 |
|-----|-----|-----|-----|
| 11^2| 53^2| 93^2| 61^2|
|-----|-----|-----|-----|
The magic constant is 15300 = a(2).

Links

Crossrefs

Cf. A016754, A271579. Subsequence of A271580. Supersequence of A271583.

A271712 a(n) = smallest magic sum of any 4 X 4 magic square composed of n-gonal numbers.

Original entry on oeis.org

1402, 2823, 2591234, 4347245, 95803378, 4818162, 160959761, 156127479
Offset: 3

Views

Author

Arkadiusz Wesolowski, Apr 12 2016

Keywords

Links

Crossrefs

Cf. A271579, A271580, A271711.

Formula

a(n) = Sum_{k=1..4} A271711(n,k).

Extensions

a(6)-a(10) were found in 2013 and 2014 by Lee Morgenstern
Showing 1-5 of 5 results.

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