A257339 -id:A257339 - cp's OEIS frontend

A257457 Indices of primes in A257339.

2, 3, 4, 6, 8, 10, 11, 12, 14, 15, 18, 19, 20, 22, 23, 27, 30, 35, 38, 42, 47, 48, 49, 52, 53, 54, 60, 61, 63, 64, 69, 70, 73, 74, 83, 85, 87, 89, 92, 94, 97, 102, 112, 114, 116, 117, 120, 125, 127, 128, 134, 136, 139, 146, 149, 151, 153, 155, 156, 158, 159

Offset: 1

Reinhard Zumkeller, Apr 24 2015

nonn

A010051(a(n)) = 1.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A257339, A010051, A257458.

a257457 n = a257457_list !! (n-1)
a257457_list = filter ((== 1) . a010051 . a257339) [1..]

A257458 Indices of prime powers in A257339.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 8, 9, 10, 11, 12, 13, 14, 15, 17, 18, 19, 20, 22, 23, 25, 27, 30, 31, 34, 35, 38, 42, 45, 47, 48, 49, 52, 53, 54, 60, 61, 63, 64, 66, 69, 70, 71, 73, 74, 81, 83, 85, 87, 89, 92, 94, 97, 100, 102, 108, 112, 114, 116, 117, 120, 125, 127, 128

Offset: 1

Reinhard Zumkeller, Apr 24 2015

nonn

A010055(a(n)) = 1.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A257339, A010055, A257457.

a257458 n = a257458_list !! (n-1)
a257458_list = filter ((== 1) . a010055 . a257339) [1..]

A257111 First differences of A257339.

Original entry on oeis.org

1, 1, 2, -1, 3, -1, 5, -2, 4, 4, 2, -11, 15, 6, -8, 4, 6, 6, 4, -6, 8, 4, -37, 39, -37, 41, -39, 1, 44, -32, 28, -16, -23, 45, -39, 29, 16, -49, 47, -41, 47, -45, 7, -1, 25, 16, 6, 4, -6, 8, 4, 8, 4, -73, 67, -61, 57, -53, 65, 4, -62, 64, 4, -93, 101, -6, 4

Offset: 1

Reinhard Zumkeller, Apr 24 2015

sign

a(n) = A257339(n+1) - A257339(n).

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A257339.

a257111 n = a257111_list !! (n-1)
a257111_list = zipWith (-) (tail a257339_list) a257339_list

A257455 Smallest m such that A257339(m) = n.

Original entry on oeis.org

1, 2, 3, 5, 4, 7, 6, 13, 9, 24, 8, 26, 10, 28, 29, 34, 11, 39, 12, 65, 16, 36, 14, 41, 17, 43, 31, 55, 15, 86, 18, 45, 44, 57, 21, 84, 19, 59, 33, 78, 20, 243, 22, 98, 62, 82, 23, 113, 25, 80, 37, 96, 27, 115, 32, 129, 46, 104, 30, 164, 35, 106, 72, 108, 40

Offset: 1

Reinhard Zumkeller, Apr 24 2015

nonn

A257339(a(n)) = n.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A257339.

import Data.List (elemIndex); import Data.Maybe (fromJust)
a257455 = (+ 1) . fromJust . (`elemIndex` a257339_list)

A257340 Arrange numbers in a single clockwise spiral so that each number is relatively prime to its four (N,S,E,W) neighbors.

Original entry on oeis.org

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

Offset: 1

N. J. A. Sloane, Apr 21 2015

nonn

Start with 1; always choose smallest number which has not yet appeared.

It is conjectured that every number appears.

			.     |     -4 | -3 | -2 | -1 |  0 | +1 | +2 | +3 | +4 | +5
.  ---+--------+----+----+----+----+----+----+----+----+----
.     |
.     |   +------------------------------------------------
.  +4 |   | 83   68   75   74   81   70   87   76   85  ...
.  ---+   |    +---------------------------------------+
.  +3 |   | 66 | 49   38   45   44   51   46   53   42 |
.  ---+   |    |    +-----------------------------+    |
.  +2 |   | 79 | 36 | 25   22   21   20   27   26 | 55 |
.  ---+   |    |    |    +-------------------+    |    |
.  +1 |   | 64 | 47 | 18 |  7    8    9   10 | 29 | 48 |
.  ---+   |    |    |    |    +---------+    |    |    |
.   0 |   | 77 | 30 | 23 |  6 |  1    2 | 11 | 24 | 59 |
.  ---+   |    |    |    |    +----o    |    |    |    |
.  -1 |   | 60 | 43 | 16 |  5    4    3 | 13 | 31 | 50 |
.  ---+   |    |    |    +--------------+    |    |    |
.  -2 |   | 73 | 40 | 19   14   15   17   12 | 35 | 57 |
.  ---+   |    |    +------------------------+    |    |
.  -3 |   | 62 | 39   34   33   41   28   37   32 | 61 |
.  ---+   |    +----------------------------------+    |
.  -4 |   | 71   56   69   67   54   65   58   63   52 |
.  ---+   +--------------------------------------------+
.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A064413, A257321-A257339.

More terms from Jon E. Schoenfield, Apr 23 2015

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

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.

A354434 a(1) = 1; for n > 1, a(n) is the smallest unused square spiral number such that a(n) shares a factor with all existing numbers in its Moore neighborhood.

Original entry on oeis.org

1, 2, 4, 6, 3, 9, 12, 18, 8, 10, 14, 16, 20, 22, 24, 15, 21, 27, 30, 33, 36, 39, 26, 28, 32, 34, 38, 40, 42, 44, 46, 48, 50, 54, 45, 35, 7, 63, 51, 57, 60, 66, 52, 72, 78, 84, 56, 58, 62, 64, 68, 70, 74, 76, 80, 82, 86, 88, 90, 75, 96, 100, 105, 49, 77, 91, 119, 102, 69, 81, 108, 92, 94, 98, 104

Offset: 1

Scott R. Shannon, May 28 2022

The sequence is conjectured to be a permutation of the positive integers, although it takes many terms for most primes to appear, e.g. a(1807) = 13, a(35156) = 179. The primes do not occur in their natural order. In the first 200000 terms the smallest unused prime is 181, while the smallest unused composite number is 11881, which is itself a prime power.

			The spiral begins
                                .
                                .
    7--35--45--54--50--48--46  82
    |                       |   |
   63  21--15--24--22--20  44  80
    |   |               |   |   |
   51  27   3---6---4  16  42  76
    |   |   |       |   |   |   |
   57  30   9   1---2  14  40  74
    |   |   |           |   |   |
   60  33  12--18---8--10  38  70
    |   |                   |   |
   66  36--39--26--28--32--34  68
    |                           |
   52--72--78--84--56--58--62--64
.
.
a(11) = 14 as the existing numbers in the Moore neighborhood when a(11) is being placed are 4,2,8,10, and 14 is the smallest unused number that shares a factor with all these numbers.

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

Cf. A336946, A064413, A253279, A257112, A257339.

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

Scott R. Shannon, May 29 2022

nonn

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.

			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.

Scott R. Shannon, Image of the first 1000 terms. The green line is y = n.

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

Scott R. Shannon, May 30 2022

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.

			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.

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

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

Scott R. Shannon, May 31 2022

nonn

See A354453 for further details.

			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.

Cf. A354453, A337116, A354441, A257339, A354434, A000040.

cp's OEIS Frontend

A257457 Indices of primes in A257339.

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A257458 Indices of prime powers in A257339.

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A257111 First differences of A257339.

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A257455 Smallest m such that A257339(m) = n.

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A257340 Arrange numbers in a single clockwise spiral so that each number is relatively prime to its four (N,S,E,W) neighbors.

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Extensions

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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A354434 a(1) = 1; for n > 1, a(n) is the smallest unused square spiral number such that a(n) shares a factor with all existing numbers in its Moore neighborhood.

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

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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.

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

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