A257252 -id:A257252 - cp's OEIS frontend

A036689 Product of a prime and the previous number.

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

A257251 Square array A(row,col) = A083221(row,col+1) - A083221(row,col): the first differences of each row of array constructed from the sieve of Eratosthenes.

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Apr 19 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:
     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
    20,  10,  20,  10,  20,  10,  20,  10,  20,  10,  20,  10,  20,  10,  20
    42,  28,  14,  28,  14,  28,  42,  14,  42,  28,  14,  28,  14,  28,  42
   110,  22,  44,  22,  44,  66,  22,  66,  44,  22,  44,  66,  66,  22,  66
   156,  52,  26,  52,  78,  26,  78,  52,  26,  52,  78,  78,  26,  78,  52
   272,  34,  68, 102,  34, 102,  68,  34,  68, 102, 102,  34, 102,  68,  34
   342,  76, 114,  38, 114,  76,  38,  76, 114, 114,  38, 114,  76,  38, 114
   506, 138,  46, 138,  92,  46,  92, 138, 138,  46, 138,  92,  46, 138,  92
   812,  58, 174, 116,  58, 116, 174, 174,  58, 174, 116,  58, 174, 116, 174
   930, 186, 124,  62, 124, 186, 186,  62, 186, 124,  62, 186, 124, 186, 248
  1332, 148,  74, 148, 222, 222,  74, 222, 148,  74, 222, 148, 222, 296, 148
  1640,  82, 164, 246, 246,  82, 246, 164,  82, 246, 164, 246, 328, 164,  82
  1806, 172, 258, 258,  86, 258, 172,  86, 258, 172, 258, 344, 172,  86, 172
  2162, 282, 282,  94, 282, 188,  94, 282, 188, 282, 376, 188,  94, 188,  94
  2756, 318, 106, 318, 212, 106, 318, 212, 318, 424, 212, 106, 212, 106, 212
  ...

Transpose: A257252.
Column 1: A036689.
Row 4: 7 * A145011.
Cf. A083221, A257253 (same array but with terms divided by 2).
Cf. arrays A257255 and A257257, also A257513.

lim = 13; Clear[row]; row[n_] := row[n] = Take[Prime[n]*Select[Range[lim^2], GCD[#*Prime[n], Product[Prime[i], {i, n-1}]] == 1&], lim] // Differences;
A[n_, k_] := row[n][[k]]; Table[A[n-k+1, k], {n, 1, lim-1}, {k, n, 1, -1}] // Flatten (* Jean-François Alcover, Mar 08 2016, after Michael De Vlieger in A083221 *)

(define (A257251 n) (A257251bi (A002260 n) (A004736 n)))
(define (A257251bi row col) (- (A083221bi row (+ 1 col)) (A083221bi row col))) ;; Code for A083221bi given in A083221.

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

A(row,col) = 2*A257253(row,col).

A257254 Transpose of square array A257253.

Original entry on oeis.org

1, 3, 1, 10, 3, 1, 21, 5, 3, 1, 55, 14, 10, 3, 1, 78, 11, 7, 5, 3, 1, 136, 26, 22, 14, 10, 3, 1, 171, 17, 13, 11, 7, 5, 3, 1, 253, 38, 34, 26, 22, 14, 10, 3, 1, 406, 69, 57, 51, 39, 33, 21, 5, 3, 1, 465, 29, 23, 19, 17, 13, 11, 7, 10, 3, 1, 666, 93, 87, 69, 57, 51, 39, 33, 21, 5, 3, 1

Offset: 1

Antti Karttunen, Apr 29 2015

See A257253.

			The top left corner of the array:
  1, 3, 10, 21, 55, 78, 136, 171, 253, 406, 465, 666, 820, 903
  1, 3,  5, 14, 11, 26,  17,  38,  69,  29,  93,  74,  41,  86
  1, 3, 10,  7, 22, 13,  34,  57,  23,  87,  62,  37,  82, 129
  1, 3,  5, 14, 11, 26,  51,  19,  69,  58,  31,  74, 123, 129
  1, 3, 10,  7, 22, 39,  17,  57,  46,  29,  62, 111, 123,  43
  1, 3,  5, 14, 33, 13,  51,  38,  23,  58,  93, 111,  41, 129
  1, 3, 10, 21, 11, 39,  34,  19,  46,  87,  93,  37, 123,  86
  1, 3,  5,  7, 33, 26,  17,  38,  69,  87,  31, 111,  82,  43
  1, 3, 10, 21, 22, 13,  34,  57,  69,  29,  93,  74,  41, 129
  1, 3,  5, 14, 11, 26,  51,  57,  23,  87,  62,  37, 123,  86
  1, 3, 10,  7, 22, 39,  51,  19,  69,  58,  31, 111,  82, 129
  1, 3,  5, 14, 33, 39,  17,  57,  46,  29,  93,  74, 123, 172
  1, 3, 10,  7, 33, 13,  51,  38,  23,  87,  62, 111, 164,  86
  1, 3,  5, 14, 11, 39,  34,  19,  69,  58,  93, 148,  82,  43
  1, 3, 10, 21, 33, 26,  17,  57,  46,  87, 124,  74,  41,  86
  1, 3,  5,  7, 22, 13,  51,  38,  69, 116,  62,  37,  82,  43
  ...

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

Transpose: A257253.
Row 1: A008837.
Cf. A083140, A083221, A257252 (same array but with terms multiplied by 2).

(define (A257254 n) (A257253bi (A004736 n) (A002260 n))) ;; Code for A257253bi given in A257253.

A257514 Transpose of square array A257513.

Original entry on oeis.org

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

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), ...

See A257513.

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

Transpose of A257513.
Column 1: A016813.
Row 1: A001223, Row 2: A069482, Row 3: A109805, Row 4: A226502 (apart from the first term).
Cf. also A083140, A083221, A257252.

(define (A257514 n) (A257513bi (A004736 n) (A002260 n))) ;; A257513bi given in A257513.

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A036689 Product of a prime and the previous number.

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A257251 Square array A(row,col) = A083221(row,col+1) - A083221(row,col): the first differences of each row of array constructed from the sieve of Eratosthenes.

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A257254 Transpose of square array A257253.

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A257514 Transpose of square array A257513.

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