A354442 -id:A354442 - cp's OEIS frontend

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

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

Scott R. Shannon, May 29 2022

nonn

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

			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.

Scott R. Shannon, Image of the first 4000 terms. The green line is y = n.

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

A354435 Lexicographically earliest sequence of distinct positive integers on a square spiral such that any 3 X 3 square of numbers sums to a prime, and these primes are distinct.

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, 39, 23, 24, 29, 36, 30, 27, 28, 34, 35, 48, 31, 32, 33, 42, 40, 41, 37, 38, 43, 44, 45, 54, 46, 49, 47, 50, 60, 63, 67, 53, 51, 52, 55, 59, 72, 75, 65, 68, 81, 56, 57, 58, 74, 85, 61, 86, 73, 62, 64, 66, 90, 87

Offset: 1

Scott R. Shannon, May 28 2022

This sequence uses the same rules as A354453 but here the sum is over every 3 X 3 square of numbers. The terms are widely spread out as in A354453 but here they display an unusual concentration in density along at least three bands that wander between the upper and lower bounds of the terms. See the linked images. The reason for this behavior is unknown.

See A354461 for the successive prime sums formed by each completed 3 X 3 square of numbers.

			The spiral begins
                                .
                                .
   32--31--48--35--34--28--27  51
    |                       |   |
   33  15--16--20--14--12  30  53
    |   |               |   |   |
   42  17   5---4---3  13  36  67
    |   |   |       |   |   |   |
   40  19   6   1---2  10  29  63
    |   |   |           |   |   |
   41  22   7---8--11---9  24  60
    |   |                   |   |
   37  18--21--25--26--39--23  50
    |                           |
   38--43--44--45--54--46--49--47
.
.
a(9) = 11 as this completes a 3 X 3 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 that has not occurred before.
a(25) = 39 as this completes a 3 X 3 square of numbers 1,2,10,8,11,9,25,26,39 which sum to 131, a prime, and 39 is the smallest unused number to form a prime sum that has not occurred before. Note that 35 would generate a square sum of 127, also a prime, but 127 was formed previously by the 3 X 3 square 19,6,1,22,7,8,18,21,25 so cannot be used. This is the first term to differ from A354441.

Scott R. Shannon, Image of the first 200000 terms. The green line is y = n.
Scott R. Shannon, Image of the first 1000000 terms.

Cf. A354461, A354441, A354442, A354453, A337116, A000040.

A354461 The primes sums formed for each completed 3 X 3 square of numbers in A354435.

Original entry on oeis.org

47, 61, 79, 71, 103, 89, 127, 107, 131, 173, 137, 149, 197, 163, 179, 191, 239, 193, 199, 211, 271, 233, 241, 263, 281, 347, 307, 311, 313, 317, 367, 331, 349, 379, 373, 389, 431, 359, 383, 401, 409, 419, 487, 421, 439, 461, 467, 479, 509, 569, 499, 503, 541, 523, 521, 547, 647, 563, 577, 593, 617

Offset: 1

Scott R. Shannon, May 31 2022

nonn

See A354435 for further details.

			The first prime sum is 47, which is the sum of the innermost 3 X 3 square of numbers 5,4,3,6,1,2,7,8,11 in the square spiral shown in A354435.
The ninth prime sum is 131, which is the sum of the 3 X 3 square of numbers 1,2,10,8,11,9,25,26,39 in the square spiral shown in A354435. This is the first sum to differ from A354442.

Cf. A354435, A354442, A354441, A354453, A337116, A000040.

cp's OEIS Frontend

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

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A354435 Lexicographically earliest sequence of distinct positive integers on a square spiral such that any 3 X 3 square of numbers sums to a prime, and these primes are distinct.

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A354461 The primes sums formed for each completed 3 X 3 square of numbers in A354435.

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