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 1-7 of 7 results.

A271579 Magic sums of 4 X 4 magic squares composed of triangular numbers.

Original entry on oeis.org

1402, 1912, 2162, 2164, 2422, 2612, 3362, 3512, 3737, 3952, 4207, 4257, 4322, 4462, 4887, 4972, 5312, 5362, 5442, 5466, 5482, 5593, 5986, 5992, 6162, 6337, 6438, 6532, 6662, 6747, 6792, 6887, 6974, 6987, 7112, 7177, 7437, 7507, 7662, 7975, 8032, 8222
Offset: 1

Views

Author

Arkadiusz Wesolowski, Apr 10 2016

Keywords

Examples

			Examples of 4 X 4 magic squares composed of triangular numbers.
.
|----|----|----|----|
| 66 |  1 | 300|1035|
|----|----|----|----|
| 465| 630| 171| 136|
|----|----|----|----|
| 780| 105| 496| 21 |
|----|----|----|----|
| 91 | 666| 435| 210|
|----|----|----|----|
The magic constant is 1402 = a(1).
.
|----|----|----|----|
|  1 | 45 | 435|1431|
|----|----|----|----|
| 300|1326| 276| 10 |
|----|----|----|----|
|1596| 190| 120|  6 |
|----|----|----|----|
| 15 | 351|1081| 465|
|----|----|----|----|
The magic constant is 1912 = a(2).

Links

Crossrefs

Cf. A000217, A271580, A271711. Subsequence of A271581.

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.

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

A358445 Euler's 4 X 4 magic square of squares.

Original entry on oeis.org

4624, 841, 1681, 1369, 289, 961, 6241, 1024, 3481, 784, 529, 3721, 121, 5929, 64, 2401
Offset: 1

Views

Author

Robert C. Lyons, Dec 30 2022

Keywords

Comments

This is the first known magic square of squares; Euler sent it to Lagrange in 1770.
The magic sum is 8515.

Examples

			The magic square is:
 [ 4624  841 1681 1369 ]
 [  289  961 6241 1024 ]
 [ 3481  784  529 3721 ]
 [  121 5929   64 2401 ]
Equivalently:
 [ 68^2 29^2 41^2 37^2 ]
 [ 17^2 31^2 79^2 32^2 ]
 [ 59^2 28^2 23^2 61^2 ]
 [ 11^2 77^2  8^2 49^2 ]

Links

Crossrefs

Cf. A271580: magic sums of 4 X 4 magic squares composed of squares.

A359383 Allan W. Johnson, Jr.'s 4 X 4 magic square of squares.

Original entry on oeis.org

900, 60516, 29584, 2025, 8649, 13456, 4356, 66564, 15876, 19044, 56169, 1936, 67600, 9, 2916, 22500
Offset: 1

Views

Author

Robert C. Lyons, Dec 28 2022

Keywords

Comments

The magic sum is 93025.

Examples

			The magic square is:
 [   900 60516 29584  2025 ]
 [  8649 13456  4356 66564 ]
 [ 15876 19044 56169  1936 ]
 [ 67600     9  2916 22500 ]
Equivalently:
 [  30^2 246^2 172^2  45^2 ]
 [  93^2 116^2  66^2 258^2 ]
 [ 126^2 138^2 237^2  44^2 ]
 [ 260^2   3^2  54^2 150^2 ]

References

  • Allan W. Johnson, Jr., A Square Magic Square, Journal of Recreational Mathematics, vol. 22:1, 1990, p. 38.

Links

Crossrefs

Cf. A271580: magic sums of 4 X 4 magic squares composed of squares.

A271583 Magic sums of 4 X 4 magic squares composed of squares of primes.

Original entry on oeis.org

509020, 736300, 860932, 1368340, 2431780
Offset: 1

Views

Author

Arkadiusz Wesolowski, Apr 10 2016

Keywords

Examples

			Examples of 4 X 4 magic squares composed of squares of primes.
.
|-----|-----|-----|-----|
| 29^2| 71^2|277^2|653^2|
|-----|-----|-----|-----|
|191^2|647^2|211^2| 97^2|
|-----|-----|-----|-----|
|673^2|139^2|163^2|101^2|
|-----|-----|-----|-----|
|137^2|257^2|601^2|251^2|
|-----|-----|-----|-----|
The magic constant is 509020 = a(1).
.
|-----|-----|-----|-----|
|179^2|107^2|383^2|739^2|
|-----|-----|-----|-----|
|353^2|691^2|331^2|157^2|
|-----|-----|-----|-----|
|761^2|283^2|263^2| 89^2|
|-----|-----|-----|-----|
| 23^2|409^2|641^2|397^2|
|-----|-----|-----|-----|
The magic constant is 736300 = a(2).

Links

Crossrefs

Cf. A271580. Subsequence of A271582.
Showing 1-7 of 7 results.

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