A257253 -id:A257253 - cp's OEIS frontend

A257254 Transpose of square array A257253.

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.

A008837 a(n) = p*(p-1)/2 for p = prime(n).

Original entry on oeis.org

1, 3, 10, 21, 55, 78, 136, 171, 253, 406, 465, 666, 820, 903, 1081, 1378, 1711, 1830, 2211, 2485, 2628, 3081, 3403, 3916, 4656, 5050, 5253, 5671, 5886, 6328, 8001, 8515, 9316, 9591, 11026, 11325, 12246, 13203, 13861, 14878, 15931, 16290, 18145, 18528, 19306

Offset: 1

N. J. A. Sloane, Ken Levasseur

Whereas A034953 is the sequence of triangular numbers with prime indices, this is the sequence of triangular numbers with numbers one less than primes for indices. - Alonso del Arte, Aug 17 2014
From Jianing Song, Apr 13 2019: (Start)
a(n) is both the number of quadratic residues and the number of nonresidues modulo prime(n)^2 that are coprime to prime(n).
For k coprime to prime(n), k^a(n) == +-1 (mod prime(n)^2). (End)

Harry J. Smith, Table of n, a(n) for n = 1..1000

Half the terms of A036689.
Cf. A000217 (triangular numbers), A112456 (least triangular number divisible by n-th prime). - Klaus Brockhaus, Nov 18 2008
Cf. A000010, A000040, A072230, A034953, A271780.
Column 1 of A257253. (Row 1 of A257254).

[ (k-1)*k/2 where k is NthPrime(n): n in [1..44] ]; // Klaus Brockhaus, Nov 18 2008

a:= n-> (p-> p*(p-1)/2)(ithprime(n)):
seq(a(n), n=1..65);  # Alois P. Heinz, Apr 20 2022

Table[Prime[n] * (Prime[n] - 1)/2, {n, 22}] (* Vladimir Joseph Stephan Orlovsky, Apr 29 2008 *)
Table[Binomial[Prime[n], 2], {n, 40}] (* Alonso del Arte, Aug 22 2014, based on the formula from Enrique Pérez Herrero *)
(#(#-1))/2&/@Prime[Range[50]] (* Harvey P. Dale, Oct 02 2019 *)

{ n=0; forprime (p=2, prime(1000), write("b008837.txt", n++, " ", p*(p - 1)/2) ) } \\ Harry J. Smith, Jul 25 2009

(define (A008837 n) (/ (A036689 n) 2)) ;; Antti Karttunen, May 01 2015

a(n) = binomial(prime(n), 2) = A000217(A000040(n)-1). - Enrique Pérez Herrero, Dec 10 2011
a(n) = (1/2)*A072230(A000040(n)). - L. Edson Jeffery, Apr 07 2012
a(n) = (phi(prime(n))^2 + phi(prime(n)))/2, where phi(n) is Euler's totient function, A000010. - Alonso del Arte, Aug 22 2014
a(n) = A036689(n)/2. - Antti Karttunen, May 01 2015
Product_{n>=2} (1 - 1/a(n)) = A271780. - Amiram Eldar, Nov 22 2022

Offset changed from 2 to 1 by Harry J. Smith, Jul 25 2009

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).

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A257254 Transpose of square array A257253.

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A008837 a(n) = p*(p-1)/2 for p = prime(n).

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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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