A162345 -id:A162345 - cp's OEIS frontend

A052288 First differences of the average of two consecutive primes (A024675).

2, 3, 3, 3, 3, 3, 5, 4, 4, 5, 3, 3, 5, 6, 4, 4, 5, 3, 4, 5, 5, 7, 6, 3, 3, 3, 3, 9, 9, 5, 4, 6, 6, 4, 6, 5, 5, 6, 4, 6, 6, 3, 3, 7, 12, 8, 3, 3, 5, 4, 6, 8, 6, 6, 4, 4, 5, 3, 6, 12, 9, 3, 3, 9, 10, 8, 6, 3, 5, 7, 7, 6, 5, 5, 7, 6, 6, 9, 6, 6, 6, 4, 5, 5, 7, 6, 3, 3, 8, 10, 6, 6, 6, 5, 9, 7, 10, 12, 8, 8, 6

Offset: 1

Labos Elemer, Feb 08 2000

			a(30) = ((113 + 127)/2) - ((127 + 131)/2) = (131 - 113)/2 = 9;
a(31) = ((127 + 131)/2) - ((137 + 131)/2) = (137 - 127)/2 = 5.

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Differences@ ListConvolve[{1, 1}/2, Prime@ Range[2, 120]] (* Michael De Vlieger, Dec 17 2016, after Jean-François Alcover at A024675 *)
Differences[Mean/@Partition[Prime[Range[2,110]],2,1]] (* Harvey P. Dale, Sep 10 2024 *)

a(n) = (prime(n+3) - prime(n+1))/2.

a(n) = A115061(n+2) = A162345(n+2). - Nathaniel Johnston, Jun 25 2011

A162343 Array read by rows in which row n lists the numbers that are in the same "y" level in the mountain path of the primes.

Original entry on oeis.org

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

Offset: 0

Omar E. Pol, Jul 24 2009

See the illustration "The mountain path of the primes", here.

			Array begins:
0, 1, 2,. . . . . . . . . . . . . . . . . .
. . . 3,. . . . . . . . . . . . . . . . . .
. . . 4, 5, 6,. . . . . . . . . . . . . . .
. . . . . . 7,. . . . . . . . . . . . . . .
. . . . . . 8,11,12,. . . . . . . . . . . .
. . . . . . 9,10,13,. . . . . . . . . . . .
. . . . . . . . .14,17,18,25,26,. . . . . .
. . . . . . . . .15,16,19,24,27,. . . . . .
. . . . . . . . . . . .20,23,28,33,34,. . .
. . . . . . . . . . . .21,22,29,32,35,. . .
. . . . . . . . . . . . . . ,30,31,36,. . .
. . . . . . . . . . . . . . . . . .37,. . .

Omar E. Pol, Graph of the mountain path function for prime numbers
Omar E. Pol, Illustration: The mountain path of the primes

Cf. A162200, A162201, A162202, A162203, A162340, A162341, A162342, A162344, A162345, A162350, A162351, A162352

A162344 Array read by rows in which row n lists the numbers that are in the same "x" level in the mountain path of the primes.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 10, 12, 13, 14, 15, 17, 16, 18, 19, 20, 21, 25, 24, 23, 22, 26, 27, 28, 29, 30, 33, 32, 31, 34, 35, 36, 37, 38, 39, 41, 40, 42, 43, 44, 45, 49, 48, 47, 46, 50, 51, 52, 53, 54, 55, 56, 59, 58, 57, 60, 61, 62, 63, 64, 68, 67, 66, 65, 69, 70, 71, 72

Offset: 0

Omar E. Pol, Jul 24 2009

See the illustration "The mountain path of the primes", here.

			Array begins:
0, 1, 2,. . . . . . . . . . . . . . . . . .
. . . 3,. . . . . . . . . . . . . . . . . .
. . . 4, 5, 6,. . . . . . . . . . . . . . .
. . . . . . 7,. . . . . . . . . . . . . . .
. . . . . . 8,11,12,. . . . . . . . . . . .
. . . . . . 9,10,13,. . . . . . . . . . . .
. . . . . . . . .14,17,18,25,26,. . . . . .
. . . . . . . . .15,16,19,24,27,. . . . . .
. . . . . . . . . . . .20,23,28,33,34,. . .
. . . . . . . . . . . .21,22,29,32,35,. . .
. . . . . . . . . . . . . . ,30,31,36,. . .
. . . . . . . . . . . . . . . . . .37,. . .

Omar E. Pol, Graph of the mountain path function for prime numbers
Omar E. Pol, Illustration: The mountain path of the primes

Cf. A162200, A162201, A162202, A162203, A162340, A162341, A162342, A162343, A162345, A162350, A162351, A162352.

A162800 a(n) = n-th grid point that is covered by the zig-zag function for prime numbers such that the grid point is a vertex in the graph of the function.

Original entry on oeis.org

0, 2, 4, 6, 9, 12, 15, 18, 21, 26, 30, 34, 39, 42, 45, 50, 56, 60, 64, 69, 72, 76, 81, 86, 93, 99, 102, 105, 108, 111, 120, 129, 134, 138, 144, 150, 154, 160, 165, 170, 176, 180, 186, 192, 195, 198, 205, 217, 225, 228, 231, 236, 240, 246, 254, 260, 266, 270, 274, 279

Offset: 0

Omar E. Pol, Jul 16 2009

Also {0, 2} together the numbers A024675.

See A162345 for the first differences.

Paolo Xausa, Table of n, a(n) for n = 0..10000

Cf. A000040, A006005, A024675, A058296, A079424, A162345, A162801, A162802.

Join[{0, 2}, Most[#] + Differences[#]/2] & [Prime[Range[2, 100]]] (* Paolo Xausa, Jun 17 2024 *)

Edited by Omar E. Pol, Jul 18 2009

A115061 a(n) is the number of occurrences of the n-th prime number in A051697.

Original entry on oeis.org

3, 2, 2, 3, 3, 3, 3, 3, 5, 4, 4, 5, 3, 3, 5, 6, 4, 4, 5, 3, 4, 5, 5, 7, 6, 3, 3, 3, 3, 9, 9, 5, 4, 6, 6, 4, 6, 5, 5, 6, 4, 6, 6, 3, 3, 7, 12, 8, 3, 3, 5, 4, 6, 8, 6, 6, 4, 4, 5, 3, 6, 12, 9, 3, 3, 9, 10, 8, 6, 3, 5, 7, 7, 6, 5, 5, 7, 6, 6

Offset: 1

Lekraj Beedassy, Mar 01 2006

Except for the second entry, the sequence also holds with respect to A077018.

a(n) equals A162345(n) for n>1 and equals A052288(n-2) for n>2. - Nathaniel Johnston, Jun 25 2011

			The 5th prime number, 11, appears three times in A051697. Therefore a(5) = 3.

Nathaniel Johnston, Table of n, a(n) for n = 1..10000

a = {3}; For[n = 2, n < 100, n++, c = 0; For[j = Prime[n - 1], j < Prime[n + 1], j++, If[j < Prime[n], If[Prime[n] - j < j - Prime[n - 1], c++ ], If[Not[Prime[n + 1] - j < j - Prime[n]], c++ ]]]; AppendTo[a, c]]; a

a(n) = (prime(n+1) - prime(n-1))/2 for n>=3. - Nathaniel Johnston, Jun 25 2011

Edited and extended by Stefan Steinerberger, Oct 27 2007

A162801 Bisection of A162800.

Original entry on oeis.org

0, 4, 9, 15, 21, 30, 39, 45, 56, 64, 72, 81, 93, 102, 108, 120, 134, 144, 154, 165, 176, 186, 195, 205, 225, 231, 240, 254, 266, 274, 282, 300, 312, 324, 342, 351, 363, 376, 386, 399, 414, 426, 436, 446, 459, 465, 483, 495, 506, 522, 544, 560, 570, 582, 596

Offset: 1

Omar E. Pol, Jul 16 2009

Essentially the same as A058296.

Cf. A058296, A079424, A162345, A162802.

A162802 Bisection of A162800.

Original entry on oeis.org

2, 6, 12, 18, 26, 34, 42, 50, 60, 69, 76, 86, 99, 105, 111, 129, 138, 150, 160, 170, 180, 192, 198, 217, 228, 236, 246, 260, 270, 279, 288, 309, 315, 334, 348, 356, 370, 381, 393, 405, 420, 432, 441, 453, 462, 473, 489, 501, 515, 532, 552, 566, 574, 590, 600

Offset: 1

Omar E. Pol, Jul 16 2009

Also, 2 together with the numbers A079424.

Cf. A079424, A162345, A162800, A162801.

cp's OEIS Frontend

A052288 First differences of the average of two consecutive primes (A024675).

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A162343 Array read by rows in which row n lists the numbers that are in the same "y" level in the mountain path of the primes.

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A162344 Array read by rows in which row n lists the numbers that are in the same "x" level in the mountain path of the primes.

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A162800 a(n) = n-th grid point that is covered by the zig-zag function for prime numbers such that the grid point is a vertex in the graph of the function.

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A115061 a(n) is the number of occurrences of the n-th prime number in A051697.

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A162801 Bisection of A162800.

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A162802 Bisection of A162800.

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