A257251 -id:A257251 - cp's OEIS frontend

A257252 Transpose of square array A257251.

2, 6, 2, 20, 6, 2, 42, 10, 6, 2, 110, 28, 20, 6, 2, 156, 22, 14, 10, 6, 2, 272, 52, 44, 28, 20, 6, 2, 342, 34, 26, 22, 14, 10, 6, 2, 506, 76, 68, 52, 44, 28, 20, 6, 2, 812, 138, 114, 102, 78, 66, 42, 10, 6, 2, 930, 58, 46, 38, 34, 26, 22, 14, 20, 6, 2, 1332, 186, 174, 138, 114, 102, 78, 66, 42, 10, 6, 2

Offset: 1

Antti Karttunen, Apr 29 2015

See A257251.

			The top left corner of the array:
  2, 6, 20, 42, 110, 156, 272, 342, 506, 812, 930, 1332
  2, 6, 10, 28,  22,  52,  34,  76, 138,  58, 186,  148
  2, 6, 20, 14,  44,  26,  68, 114,  46, 174, 124,   74
  2, 6, 10, 28,  22,  52, 102,  38, 138, 116,  62,  148
  2, 6, 20, 14,  44,  78,  34, 114,  92,  58, 124,  222
  2, 6, 10, 28,  66,  26, 102,  76,  46, 116, 186,  222
  2, 6, 20, 42,  22,  78,  68,  38,  92, 174, 186,   74
  2, 6, 10, 14,  66,  52,  34,  76, 138, 174,  62,  222
  2, 6, 20, 42,  44,  26,  68, 114, 138,  58, 186,  148
  2, 6, 10, 28,  22,  52, 102, 114,  46, 174, 124,   74
  2, 6, 20, 14,  44,  78, 102,  38, 138, 116,  62,  222
  2, 6, 10, 28,  66,  78,  34, 114,  92,  58, 186,  148
  2, 6, 20, 14,  66,  26, 102,  76,  46, 174, 124,  222
  2, 6, 10, 28,  22,  78,  68,  38, 138, 116, 186,  296
  2, 6, 20, 42,  66,  52,  34, 114,  92, 174, 248,  148
  2, 6, 10, 14,  44,  26, 102,  76, 138, 232, 124,   74
  ...

Antti Karttunen, Table of n, a(n) for n = 1..3321; the first 81 antidiagonals of the array

Transpose: A257251.
Row 1: A036689.
Cf. also A083140, A257254 (same array but with terms divided by 2).

(define (A257252 n) (A257251bi (A004736 n) (A002260 n))) ;; Other code as in A257251.

A036689 Product of a prime and the previous number.

Original entry on oeis.org

2, 6, 20, 42, 110, 156, 272, 342, 506, 812, 930, 1332, 1640, 1806, 2162, 2756, 3422, 3660, 4422, 4970, 5256, 6162, 6806, 7832, 9312, 10100, 10506, 11342, 11772, 12656, 16002, 17030, 18632, 19182, 22052, 22650, 24492, 26406, 27722, 29756, 31862, 32580, 36290, 37056, 38612, 39402, 44310

Offset: 1

Felice Russo

Records in A002618. - Artur Jasinski, Jan 23 2008

Also records in A174857. - Vladimir Shevelev, Mar 31 2010

			2*1, 3*2, 5*4, 7*6, 11*10, 13*12, 17*16, ...

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Index to sequences related to prime powers

Cf. A000040, A001248, A002618, A005596, A053650, A053192, A053193, A053650, A082695, A117495, A136141.
Twice the terms of A008837.
Subsequence of A002378 (oblong numbers).
Column 1 of A257251. (Row 1 of A257252.)
Column 2 of A379010.

a036689 n = a036689_list !! (n-1)
a036689_list = zipWith (*) a000040_list $ map pred a000040_list
-- Reinhard Zumkeller, Sep 17 2011

[ n*(n-1): n in PrimesUpTo(220) ]; // Bruno Berselli, Apr 11 2011

A036689 := proc(n) local p ; p := ithprime(n) ; p*(p-1) ; end proc: # R. J. Mathar, Apr 11 2011

Table[Prime[n] EulerPhi[Prime[n]], {n, 100}] (* Artur Jasinski, Jan 23 2008 *)
Table[Prime[n] (Prime[n] - 1), {n, 1, 50}] (* Bruno Berselli, Apr 22 2014 *)
#(#-1)&/@Prime[Range[50]] (* Harvey P. Dale, Sep 08 2019 *)

forprime(p=2,1e3,print1(p^2-p", ")) \\ Charles R Greathouse IV, Jun 10 2011

A036689(n) = ((p->(p-1)*p)(prime(n))); \\ Antti Karttunen, Dec 14 2024

(define (A036689 n) (* (A000040 n) (- (A000040 n) 1))) ;; Antti Karttunen, May 01 2015

a(n) = prime(n) * (prime(n) - 1).
a(n) = phi(prime(n)^2) = A000010(A001248(n)).
a(n) = prime(n) * phi(prime(n)). - Artur Jasinski, Jan 23 2008
From Reinhard Zumkeller, Sep 17 2011: (Start)
a(n) = A000040(n) * A006093(n) = A001248(n) - A000040(n).
A006530(a(n)) = A000040(n). (End)
a(n) = A009262(prime(n)). - Enrique Pérez Herrero, May 12 2012
a(n) = prime(n)! mod (prime(n)^2). - J. M. Bergot, Apr 10 2014
a(n) = 2*A008837(n). - Antti Karttunen, May 01 2015
Sum_{n>=1} 1/a(n) = A136141. - Amiram Eldar, Nov 09 2020
From Amiram Eldar, Jan 23 2021: (Start)
Product_{n>=1} (1 + 1/a(n)) = zeta(2)*zeta(3)/zeta(6) (A082695).
Product_{n>=1} (1 - 1/a(n)) = A005596. (End)

Deleted two incorrect comments. - N. J. A. Sloane, May 07 2020

A257257 Square array A(row,col) = A255127(row,col+1) - A255127(row,col): the first differences of each row of Ludic array.

Original entry on oeis.org

2, 2, 6, 2, 6, 14, 2, 6, 16, 24, 2, 6, 14, 28, 44, 2, 6, 16, 26, 48, 60, 2, 6, 14, 28, 48, 60, 84, 2, 6, 16, 24, 52, 64, 86, 122, 2, 6, 14, 26, 48, 66, 94, 126, 142, 2, 6, 16, 28, 48, 62, 86, 132, 144, 176, 2, 6, 14, 26, 44, 60, 94, 120, 146, 166, 216, 2, 6, 16, 24, 48, 64, 86, 132, 142, 180, 234, 252

Offset: 1

Antti Karttunen, Apr 19 2015

The array A(row,col) is read by downwards antidiagonals as A(1,1), A(1,2), A(2,1), A(1,3), A(2,2), A(3,1), ...

			The top left corner of the array:
    2,   2,   2,   2,   2,   2,   2,   2,   2,   2,   2,   2,   2,   2,   2
    6,   6,   6,   6,   6,   6,   6,   6,   6,   6,   6,   6,   6,   6,   6
   14,  16,  14,  16,  14,  16,  14,  16,  14,  16,  14,  16,  14,  16,  14
   24,  28,  26,  28,  24,  26,  28,  26,  24,  28,  26,  28,  24,  26,  28
   44,  48,  48,  52,  48,  48,  44,  48,  52,  48,  48,  48,  44,  52,  48
   60,  60,  64,  66,  62,  60,  64,  62,  60,  70,  60,  60,  62,  64,  60
   84,  86,  94,  86,  94,  86,  82,  92,  88,  92,  88,  90,  90,  84,  90
  122, 126, 132, 120, 132, 126, 130, 126, 120, 132, 128, 126, 130, 128, 126
  142, 144, 146, 142, 146, 138, 150, 148, 140, 148, 146, 138, 150, 138, 150
  176, 166, 180, 168, 176, 178, 170, 178, 170, 180, 174, 172, 176, 178, 176
  216, 234, 226, 242, 228, 226, 240, 218, 234, 246, 220, 230, 234, 226, 234
  252, 270, 254, 274, 258, 254, 258, 276, 262, 266, 258, 256, 264, 276, 264
  274, 284, 268, 284, 304, 270, 282, 278, 294, 282, 282, 276, 282, 288, 292
  308, 316, 314, 316, 320, 316, 312, 308, 324, 336, 316, 302, 316, 314, 322
  360, 360, 354, 368, 360, 372, 370, 368, 352, 360, 380, 354, 370, 380, 352
  412, 434, 424, 420, 426, 440, 426, 420, 432, 424, 422, 444, 424, 422, 430
  ...

Antti Karttunen, Table of n, a(n) for n = 1..10440; the first 144 antidiagonals of the array

Column 1: A256482.
Cf. A255127.
Cf. A257258 (same array but with terms divided by 2).
Cf. also arrays A257251 and A257255.

(define (A257257 n) (A257257bi (A002260 n) (A004736 n)))
(define (A257257bi row col) (- (A255127bi row (+ 1 col)) (A255127bi row col))) ;; Code for A255127bi given in A255127.

A(row,col) = A255127(row,col+1) - A255127(row,col).

A(row,col) = 2*A257258(row,col).

A145011 First differences of A007775.

Original entry on oeis.org

6, 4, 2, 4, 2, 4, 6, 2, 6, 4, 2, 4, 2, 4, 6, 2, 6, 4, 2, 4, 2, 4, 6, 2, 6, 4, 2, 4, 2, 4, 6, 2, 6, 4, 2, 4, 2, 4, 6, 2, 6, 4, 2, 4, 2, 4, 6, 2, 6, 4, 2, 4, 2, 4, 6, 2, 6, 4, 2, 4, 2, 4, 6, 2, 6, 4, 2, 4, 2, 4, 6, 2, 6, 4, 2, 4, 2, 4, 6, 2, 6, 4, 2, 4, 2, 4, 6, 2, 6, 4, 2, 4, 2, 4, 6, 2

Offset: 1

Ki Punches, Feb 25 2009

Also the first differences of A084968 divided by 7. - Antti Karttunen, May 01 2015

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,0,0,0,1).

Cf. A007775, A084968.

Multiplied by 7: row 4 of A257251.

a145011 n = a145011_list !! (n-1)
a145011_list = zipWith (-) (tail a007775_list) a007775_list
-- Reinhard Zumkeller, Jan 06 2013

Differences[Select[Range[400],GCD[#,30]==1&]] (* Harvey P. Dale, Dec 07 2011 *)

a(n)=[4,6,4,2,4,2][n%8+1] \\ Charles R Greathouse IV, Oct 20 2013

Period 8: repeat 6,4,2,4,2,4,6,2.
a(n) = 2*((abs(abs((n mod 8) - 3) - 1) mod 3) + 1). - Pieter Stadhouders, Mar 09 2010
G.f.: x*(-2*x^7 - 6*x^6 - 4*x^5 - 2*x^4 - 4*x^3 - 2*x^2 - 4*x - 6)/(x^8 - 1). - Chai Wah Wu, Feb 16 2021

Edited by Omar E. Pol, Mar 02 2009

Offset corrected by Reinhard Zumkeller, Jan 06 2013

A257253 Square array A(row,col) = (1/2) * (A083221(row,col+1) - A083221(row,col)): half of the first differences of each row of array constructed from the sieve of Eratosthenes.

Original entry on oeis.org

1, 1, 3, 1, 3, 10, 1, 3, 5, 21, 1, 3, 10, 14, 55, 1, 3, 5, 7, 11, 78, 1, 3, 10, 14, 22, 26, 136, 1, 3, 5, 7, 11, 13, 17, 171, 1, 3, 10, 14, 22, 26, 34, 38, 253, 1, 3, 5, 21, 33, 39, 51, 57, 69, 406, 1, 3, 10, 7, 11, 13, 17, 19, 23, 29, 465

Offset: 1

Antti Karttunen, Apr 29 2015

The array A(row,col) is read by its downwards antidiagonals as A(1,1), A(1,2), A(2,1), A(1,3), A(2,2), A(3,1), ...

			The top left corner of the array:
     1,   1,   1,   1,   1,   1,   1,   1,   1,   1,   1,   1,   1,   1,   1
     3,   3,   3,   3,   3,   3,   3,   3,   3,   3,   3,   3,   3,   3,   3
    10,   5,  10,   5,  10,   5,  10,   5,  10,   5,  10,   5,  10,   5,  10
    21,  14,   7,  14,   7,  14,  21,   7,  21,  14,   7,  14,   7,  14,  21
    55,  11,  22,  11,  22,  33,  11,  33,  22,  11,  22,  33,  33,  11,  33
    78,  26,  13,  26,  39,  13,  39,  26,  13,  26,  39,  39,  13,  39,  26
   136,  17,  34,  51,  17,  51,  34,  17,  34,  51,  51,  17,  51,  34,  17
   171,  38,  57,  19,  57,  38,  19,  38,  57,  57,  19,  57,  38,  19,  57
   253,  69,  23,  69,  46,  23,  46,  69,  69,  23,  69,  46,  23,  69,  46
   406,  29,  87,  58,  29,  58,  87,  87,  29,  87,  58,  29,  87,  58,  87
   465,  93,  62,  31,  62,  93,  93,  31,  93,  62,  31,  93,  62,  93, 124
   666,  74,  37,  74, 111, 111,  37, 111,  74,  37, 111,  74, 111, 148,  74
   820,  41,  82, 123, 123,  41, 123,  82,  41, 123,  82, 123, 164,  82,  41
   903,  86, 129, 129,  43, 129,  86,  43, 129,  86, 129, 172,  86,  43,  86
  1081, 141, 141,  47, 141,  94,  47, 141,  94, 141, 188,  94,  47,  94,  47
  1378, 159,  53, 159, 106,  53, 159, 106, 159, 212, 106,  53, 106,  53, 106
  ...

Antti Karttunen, Table of n, a(n) for n = 1..3321; the first 81 antidiagonals of the array

Transpose: A257254.
Cf. A083221, A257251 (same array but with terms multiplied by 2).
Column 1: A008837.
Row 4: (7/2) * A145011.

(define (A257253 n) (A257253bi (A002260 n) (A004736 n)))
(define (A257253bi row col) (* (/ 1 2) (- (A083221bi row (+ 1 col)) (A083221bi row col)))) ;; Code for A083221bi given in A083221.

A(row,col) = (1/2) * (A083221(row,col+1) - A083221(row,col)).

A(row,col) = A257251(row,col)/2.

A257255 Square array A(row,col) = A255545(row,col+1) - A255545(row,col): the first differences of each row of Lucky-Unlucky array.

Original entry on oeis.org

1, 2, 2, 2, 6, 12, 2, 6, 20, 18, 2, 6, 22, 30, 32, 2, 6, 20, 34, 52, 40, 2, 6, 22, 30, 50, 62, 64, 2, 6, 20, 32, 52, 64, 92, 84, 2, 6, 22, 30, 54, 62, 100, 116, 108, 2, 6, 20, 34, 48, 72, 92, 120, 156, 124, 2, 6, 22, 30, 50, 64, 102, 120, 152, 168, 138, 2, 6, 20, 32, 52, 62, 96, 124, 156, 168, 206, 170

Offset: 1

Antti Karttunen, Apr 19 2015

The array A(row,col) is read by downwards antidiagonals as A(1,1), A(1,2), A(2,1), A(1,3), A(2,2), A(3,1), ...

			The top left corner of the array:
    1,   2,   2,   2,   2,   2,   2,   2,   2,   2,   2,   2,   2,   2,   2
    2,   6,   6,   6,   6,   6,   6,   6,   6,   6,   6,   6,   6,   6,   6
   12,  20,  22,  20,  22,  20,  22,  20,  22,  20,  22,  20,  22,  20,  22
   18,  30,  34,  30,  32,  30,  34,  30,  32,  30,  34,  30,  32,  30,  34
   32,  52,  50,  52,  54,  48,  50,  52,  50,  54,  48,  54,  52,  48,  54
   40,  62,  64,  62,  72,  64,  62,  64,  66,  62,  64,  62,  66,  64,  62
   64,  92, 100,  92, 102,  96,  96,  94,  96,  96,  96,  96,  98,  94,  98
   84, 116, 120, 120, 124, 116, 124, 116, 118, 122, 120, 118, 126, 120, 120
  108, 156, 152, 156, 162, 148, 162, 152, 150, 160, 152, 154, 156, 156, 158
  124, 168, 168, 174, 168, 164, 178, 170, 166, 174, 174, 168, 176, 162, 168
  138, 206, 192, 198, 198, 190, 200, 202, 192, 200, 190, 198, 200, 192, 208
  170, 232, 236, 238, 230, 244, 230, 240, 226, 242, 238, 234, 230, 246, 222
  206, 270, 274, 278, 268, 272, 280, 278, 268, 276, 276, 282, 266, 270, 286
  214, 284, 300, 286, 302, 288, 292, 288, 290, 294, 292, 290, 298, 284, 300
  274, 366, 356, 390, 358, 372, 354, 374, 378, 360, 360, 376, 366, 372, 366
  296, 384, 418, 392, 400, 396, 398, 390, 396, 402, 394, 402, 398, 400, 392
  ...

Antti Karttunen, Table of n, a(n) for n = 1..10440; the first 144 antidiagonals of the array

Column 1: A257256.
Cf. A255545.
Cf. also arrays A257251 and A257257.

(define (A257255 n) (A257255bi (A002260 n) (A004736 n)))
(define (A257255bi row col) (- (A255545bi row (+ 1 col)) (A255545bi row col))) ;; Code for A255545bi given in A255545.

A(row,col) = A255545(row,col+1) - A255545(row,col).

A257513 Square array A(row,col) = A083221(row+1,col) - A083221(row,col): the first differences of each column of array constructed from the sieve of Eratosthenes.

Original entry on oeis.org

1, 5, 2, 9, 16, 2, 13, 20, 24, 4, 17, 34, 42, 72, 2, 21, 38, 36, 66, 48, 4, 25, 52, 54, 96, 78, 120, 2, 29, 56, 48, 90, 60, 102, 72, 4, 33, 70, 66, 120, 90, 144, 114, 168, 6, 37, 74, 88, 158, 124, 194, 160, 230, 312, 2, 41, 88, 92, 138, 84, 150, 96, 162, 232, 120, 6, 45, 92, 114, 190, 140, 226, 176, 262, 360, 248, 408, 4

Offset: 1

Antti Karttunen, May 01 2015

The array is read by downwards antidiagonals as A(1,1), A(1,2), A(2,1), A(1,3), A(2,2), A(3,1), ...

			The top left corner of the array:
1,   5,   9,  13,  17,  21,  25,  29,  33,  37,  41,  45,  49,  53,  57,  61
2,  16,  20,  34,  38,  52,  56,  70,  74,  88,  92, 106, 110, 124, 128, 142
2,  24,  42,  36,  54,  48,  66,  88,  92, 114, 132, 126, 144, 138, 156, 178
4,  72,  66,  96,  90, 120, 158, 138, 190, 192, 186, 216, 254, 306, 300, 324
2,  48,  78,  60,  90, 124,  84, 140, 126, 108, 138, 172, 184, 144, 200, 186
4, 120, 102, 144, 194, 150, 226, 216, 198, 240, 290, 314, 270, 346, 336, 318
2,  72, 114, 160,  96, 176, 150, 120, 162, 208, 220, 156, 236, 210, 180, 260
4, 168, 230, 162, 262, 240, 210, 264, 326, 350, 282, 382, 360, 330, 430, 408
6, 312, 232, 360, 338, 304, 374, 456, 492, 412, 540, 518, 484, 612, 590, 672
2, 120, 248, 198, 144, 210, 280, 292, 180, 308, 258, 204, 332, 282, 352, 426
6, 408, 370, 320, 406, 504, 540, 428, 588, 550, 500, 660, 622, 720, 830, 730
4, 312, 246, 336, 434, 458, 318, 490, 432, 366, 538, 480, 578, 684, 552, 486
2, 168, 258, 352, 364, 204, 380, 306, 228, 404, 330, 424, 522, 366, 288, 378
4, 360, 470, 494, 330, 526, 456, 378, 574, 504, 614, 732, 576, 498, 600, 522
6, 600, 636, 460, 684, 614, 532, 756, 686, 816, 958, 794, 712, 830, 748, 866
...

Transpose: A257514.
Row 1: A016813.
Column 1: A001223, Column 2: A069482, Column 3: A109805, Column 4: A226502 (apart from the first term).
Cf. A083221, A257251.

(define (A257513 n) (A257513bi (A002260 n) (A004736 n)))
(define (A257513bi row col) (- (A083221bi (+ 1 row) col) (A083221bi row col))) ;; A083221bi given in A083221.

A(row,col) = A083221(row+1,col) - A083221(row,col).

cp's OEIS Frontend

A257252 Transpose of square array A257251.

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A036689 Product of a prime and the previous number.

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A257257 Square array A(row,col) = A255127(row,col+1) - A255127(row,col): the first differences of each row of Ludic array.

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A145011 First differences of A007775.

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A257253 Square array A(row,col) = (1/2) * (A083221(row,col+1) - A083221(row,col)): half of the first differences of each row of array constructed from the sieve of Eratosthenes.

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A257255 Square array A(row,col) = A255545(row,col+1) - A255545(row,col): the first differences of each row of Lucky-Unlucky array.

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A257513 Square array A(row,col) = A083221(row+1,col) - A083221(row,col): the first differences of each column of array constructed from the sieve of Eratosthenes.

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