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 1-4 of 4 results.

A354441 Lexicographically earliest sequence of distinct positive integers on a square spiral such that any 3X3 square of numbers sums to a prime.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 11, 9, 10, 13, 12, 14, 20, 16, 15, 17, 19, 22, 18, 21, 25, 26, 35, 23, 24, 27, 28, 30, 29, 31, 33, 37, 41, 36, 32, 34, 43, 38, 40, 52, 39, 42, 66, 48, 45, 44, 46, 47, 49, 54, 50, 56, 51, 57, 53, 55, 61, 72, 67, 59, 58, 62, 60, 63, 71, 68, 74, 76, 70, 80, 64, 65, 69, 77, 73
Offset: 1

Views

Author

Scott R. Shannon, May 29 2022

Keywords

Comments

See A354442 for the successive prime sums formed by each completed 3X3 square of numbers.

Examples

			The spiral begins
                                .
                                .
   32--36--41--37--33--31--29  57
    |                       |   |
   34  15--16--20--14--12  30  51
    |   |               |   |   |
   43  17   5---4---3  13  28  56
    |   |   |       |   |   |   |
   38  19   6   1---2  10  27  50
    |   |   |           |   |   |
   40  22   7---8--11---9  24  54
    |   |                   |   |
   52  18--21--25--26--35--23  49
    |                           |
   39--42--66--48--45--44--46--47
.
.
a(9) = 11 as this completes a 3X3 square of numbers 5,4,3,6,1,2,7,8,11, which sum to 47, a prime, and 11 is the smallest unused number to form a prime sum.
a(12) = 13 as this completes a 3X3 square of numbers 8,11,9,1,2,10,4,3,13, which sum to 61, a prime, and 13 is the smallest unused number to form a prime sum.

Links

Crossrefs

Cf. A354442, A337116, A257339, A354434, A000040.

A354442 The primes sums formed for each completed 3 X 3 square of numbers in A354441.

Original entry on oeis.org

47, 61, 79, 71, 103, 89, 127, 107, 127, 167, 127, 139, 193, 167, 173, 191, 239, 193, 197, 223, 307, 257, 257, 251, 263, 331, 281, 271, 277, 307, 379, 337, 347, 359, 349, 353, 431, 379, 379, 397, 409, 439, 499, 449, 439, 463, 457, 461, 479, 569, 499, 491, 509, 521, 523, 557, 643, 557, 563, 599, 613
Offset: 1

Views

Author

Scott R. Shannon, May 29 2022

Keywords

Comments

See A354441 for further details.
In the first one million terms the most frequently occurring prime sum is 8986531, which occurs twenty-eight times. It is unknown if the maximum number of times a prime sum can occur is finite or unbounded.

Examples

			The first prime sum is 47, which is the sum of the innermost nine values 1,2,3,4,5,6,7,8,11 which form the 3 X 3 square centered at (0,0) in the square spiral shown in A354441.
The second prime sum is 61, which is the sum of the nine values 1,2,3,4,8,9,10,11,13 which form the 3 X 3 square centered at (1,0) in the square spiral shown in A354441.

Links

Crossrefs

Cf. A354441, A337116, A257339, A354434, A000040.

A354453 Lexicographically earliest sequence of distinct positive integers on a square spiral such that any 2 X 2 square of numbers sums to a prime, and that prime is unique for all such squares. Start with a(1) = 0.

Original entry on oeis.org

0, 1, 2, 4, 3, 6, 5, 8, 14, 7, 9, 17, 10, 12, 19, 21, 11, 18, 16, 32, 13, 23, 25, 20, 30, 15, 27, 40, 31, 43, 22, 28, 39, 37, 36, 41, 24, 51, 57, 48, 35, 69, 26, 49, 66, 53, 65, 58, 76, 29, 61, 88, 38, 90, 33, 113, 34, 54, 123, 67, 86, 74, 100, 98, 42, 75, 91, 70, 96, 102, 71, 117, 44, 106, 126
Offset: 1

Views

Author

Scott R. Shannon, May 30 2022

Keywords

Comments

This is a variation of A337116 where the same rules apply except that the primes generated by all 2 X 2 square sums must be unique. This leads to the terms having a far greater variation in value while being concentrated along a central line which shows wave-like variations in density. See the linked image. The reason for this behavior is unknown.
See A354460 for the successive prime sums formed by each completed 2 X 2 square of numbers.

Examples

			The spiral begins
                                .
                                .
   24--41--36--37--39--28--22 113
    |                       |   |
   51  11--21--19--12--10  43  33
    |   |               |   |   |
   57  18   3---4---2  17  31  90
    |   |   |       |   |   |   |
   48  16   6   0---1   9  40  38
    |   |   |           |   |   |
   35  32   5---8--14---7  27  88
    |   |                   |   |
   69  13--23--25--20--30--15  61
    |                           |
   26--49--66--53--65--58--76--29
.
.
a(9) = 14 as this completes a 2 X 2 square of numbers 0,1,8,14 which sum to 23, a prime, and 14 is the smallest unused number to form a prime sum that has not occurred before. Note that 10 is unused and would form a prime sum of 19, see A337116, but 19 was formed previously by the square 6,0,5,8, so cannot be used. This is the first term to differ from A337116.

Links

Crossrefs

Cf. A354460, A337116, A354441, A257339, A354434, A000040.

A354460 The primes sums formed for each completed 2 X 2 square of numbers in A354453.

Original entry on oeis.org

7, 13, 19, 23, 31, 29, 41, 37, 47, 53, 43, 59, 73, 61, 67, 71, 79, 83, 97, 101, 103, 89, 107, 113, 109, 127, 137, 139, 131, 149, 157, 151, 167, 163, 173, 179, 181, 191, 193, 199, 197, 211, 223, 227, 257, 229, 233, 251, 263, 239, 241, 269, 271, 281, 277, 283, 293, 307, 313, 311, 317, 331, 347, 337
Offset: 1

Views

Author

Scott R. Shannon, May 31 2022

Keywords

Comments

See A354453 for further details.

Examples

			The first prime sum is 7, which is the sum of the innermost 2 X 2 square of values 4,2,0,1 in the square spiral shown in A354453.
The fourth prime sum is 23, which is the sum of the 2 X 2 square of values 0,1,8,14 in the square spiral shown in A354453. This is the first prime sum that differs from A337116.

Crossrefs

Cf. A354453, A337116, A354441, A257339, A354434, A000040.
Showing 1-4 of 4 results.

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