A268729 -id:A268729 - cp's OEIS frontend

A268728 Square array A(row,col) = B(row,(2*col)-1), where B(p,2q-1) = 0 if gcd(p,2q-1) > 1, and A269158(p,q) otherwise. Array is read by descending antidiagonals as A(1,1), A(1,2), A(2,1), A(1,3), A(2,2), A(3,1), ...

0, 0, 1, 0, 2, 1, 0, 7, 0, 0, 0, 4, 3, 0, 1, 0, 13, 3, 0, 2, 0, 0, 14, 0, 0, 0, 0, 1, 0, 11, 1, 0, 2, 4, 0, 1, 0, 8, 1, 0, 1, 7, 7, 2, 1, 0, 25, 0, 0, 1, 0, 0, 7, 0, 0, 0, 26, 3, 0, 6, 15, 5, 4, 0, 0, 1, 0, 31, 3, 0, 0, 10, 3, 13, 4, 0, 2, 1, 0, 28, 0, 0, 6, 0, 2, 14, 0, 6, 0, 0, 1, 0, 21, 1, 0, 1, 26, 7, 11, 4, 12, 0, 3, 0, 0

Offset: 1

Antti Karttunen, Feb 19 2016

The array gives the values of bivariate function B(p,q) which is well-defined only when q is odd, thus while here its argument p obtains all integer values from 1 onward, argument q gets only odd numbers 1, 3, 5, 7, 9, ... as its values.

Any row n occurs also as row (4^k * n), for all k >= 0.

			The top left [1 .. 16] x [1 .. 25] section of the array:
  0, 0, 0, 0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0
  1, 2, 7, 4, 13, 14, 11,  8, 25, 26, 31, 28, 21, 22, 19, 16
  1, 0, 3, 3,  0,  1,  1,  0,  3,  3,  0,  1,  1,  0,  3,  3
  0, 0, 0, 0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0
  1, 2, 0, 2,  1,  1,  6,  0,  6,  1,  5,  6,  0,  6,  5,  5
  0, 0, 4, 7,  0, 15, 10,  0, 26, 25,  0, 29, 20,  0, 16, 19
  1, 0, 7, 0,  5,  3,  2,  7,  2,  1,  0,  3,  1,  4,  5,  4
  1, 2, 7, 4, 13, 14, 11,  8, 25, 26, 31, 28, 21, 22, 19, 16
  1, 0, 0, 4,  0,  4,  9,  0, 12,  1,  0,  0, 12,  0,  4,  9
  0, 0, 0, 6, 12, 15, 13,  0, 31, 27, 26, 26,  0, 16, 22, 21
  1, 2, 0, 0, 13,  0,  7, 11, 14, 13, 14,  3,  8, 10, 10, 15
  1, 0, 3, 3,  0,  1,  1,  0,  3,  3,  0,  1,  1,  0,  3,  3
  1, 0, 3, 7,  0, 14,  0,  6,  1, 11, 14,  8,  8,  9, 12, 11
  0, 2, 0, 0,  8, 13,  9, 15, 27, 27,  0, 31, 20, 18, 22, 20
  1, 0, 0, 0,  0,  0, 11,  0,  9,  3,  0, 15,  0,  0,  2, 15
  0, 0, 0, 0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0
  1, 2, 7, 3, 13, 15,  0,  8,  0,  8, 17, 11,  8, 14, 18, 10
  0, 0, 7, 0,  0, 10,  2,  0, 21, 27,  0, 28, 25,  0, 23, 25
  1, 0, 0, 2,  0, 14, 10,  0, 25,  0, 11, 19,  8,  9, 10, 16
  1, 2, 0, 2,  1,  1,  6,  0,  6,  1,  5,  6,  0,  6,  5,  5
  1, 0, 0, 0,  0, 15, 11,  0,  0, 26,  0, 10, 17,  0, 10, 15
  0, 0, 7, 4,  0,  0, 12,  3, 23, 23, 17, 31, 29, 28, 25, 31
  1, 2, 3, 4,  1,  0, 13,  8, 26,  0, 31,  0, 13, 19,  8, 11
  0, 0, 4, 7,  0, 15, 10,  0, 26, 25,  0, 29, 20,  0, 16, 19
  1, 0, 0, 0,  5,  1,  1,  0, 25, 25,  0, 28,  0, 12, 25, 13

Antti Karttunen, Table of n, a(n) for n = 1..32896; the first 256 antidiagonals of the array

Transpose: A268729.
Column 1: Seems to be 0 followed by A039982.
Cf. A065621 (occurs as row 2, row 8, and in general, as any row 2^(2n+1) for n >= 0).
Cf. A268829, A269158 (variants).
Cf. A003188, A003987, A004198.

(define (A268728 n) (A268728bi (A002260 n) (+ -1 (* 2 (A004736 n)))))
(define (A268728bi p q) (if (not (odd? q)) (error "A268728bi: the second argument should be odd: " p q) (let loop ((p p) (q q) (s 0)) (cond ((zero? p) 0) ((= 1 p) s) ((odd? p) (loop (modulo q p) p (A003987bi s (A004198bi p q)))) (else (loop (/ p 2) q (A003987bi s (A003987bi q (/ (- q 1) 2)))))))))
;; Alternative implementation using the definition given in A269158:
(define (A268728 n) (let ((p (A002260 n)) (q (+ -1 (* 2 (A004736 n))))) (if (< 1 (gcd p q)) 0 (A269158auxbi p q))))

A(row,col) = B(row,(2*col)-1), where function B(p,q) [only odd values allowed for q] is defined as: If gcd(p,q) > 1, B(p,q) = 0, otherwise B(p,q) = F(p,q) = A269158(p,(q+1)/2), function F defined as in A269158.

A269159 Transpose of A269158.

Original entry on oeis.org

0, 1, 0, 1, 2, 0, 0, 3, 7, 0, 1, 0, 3, 4, 0, 0, 2, 0, 3, 13, 0, 1, 1, 5, 0, 1, 14, 0, 1, 0, 4, 2, 0, 1, 11, 0, 1, 2, 7, 7, 1, 0, 1, 8, 0, 0, 0, 7, 7, 12, 1, 0, 3, 25, 0, 1, 0, 0, 4, 5, 15, 6, 0, 3, 26, 0, 1, 2, 2, 4, 13, 3, 10, 5, 0, 3, 31, 0, 1, 3, 0, 6, 9, 14, 2, 11, 6, 0, 1, 28, 0, 0, 0, 3, 0, 12, 4, 11, 7, 26, 1, 0, 1, 21, 0

Offset: 1

Antti Karttunen, Feb 20 2016

See comments in A269158.

			The top left [1 .. 19] x [1 .. 19] section of the array:
0,  1, 1, 0,  1,  0,  1,  1,  1,  0,  1,  1,  1,  0,  1,  0,  1,  0,  1
0,  2, 3, 0,  2,  1,  0,  2,  0,  0,  2,  3,  0,  2,  0,  0,  2,  2,  0
0,  7, 3, 0,  5,  4,  7,  7,  0,  2,  0,  3,  3,  0,  0,  0,  7,  7,  0
0,  4, 3, 0,  2,  7,  7,  4,  4,  6,  0,  3,  7,  3,  0,  0,  3,  0,  2
0, 13, 1, 0,  1, 12,  5, 13,  9, 12, 13,  1,  0,  8, 12,  0, 13,  4,  0
0, 14, 1, 0,  1, 15,  3, 14,  4, 15, 11,  1, 14, 13,  0,  0, 15, 10, 14
0, 11, 1, 0,  6, 10,  2, 11,  9, 13,  7,  1, 13,  9, 11,  0,  0,  2, 10
0,  8, 3, 0,  5, 11,  7,  8,  5, 13, 11,  3,  6, 15, 15,  0,  8, 13,  0
0, 25, 3, 0,  6, 26,  2, 25, 12, 31, 14,  3,  1, 27,  9,  0, 17, 21, 25
0, 26, 3, 0,  1, 25,  1, 26,  1, 27, 13,  3, 11, 27,  3,  0,  8, 27, 19
0, 31, 1, 0,  5, 30,  5, 31,  0, 26, 14,  1, 14, 26, 14,  0, 17, 31, 11
0, 28, 1, 0,  6, 29,  3, 28,  0, 26,  3,  1,  8, 31, 15,  0, 11, 28, 19
0, 21, 1, 0,  1, 20,  1, 21, 12, 20,  8,  1,  8, 20,  4,  0,  8, 25,  8
0, 22, 3, 0,  6, 21,  4, 22,  9, 16, 10,  3,  9, 18,  8,  0, 14, 31,  9
0, 19, 3, 0,  5, 16,  5, 19,  4, 22, 10,  3, 12, 22,  2,  0, 18, 23, 10
0, 16, 3, 0,  5, 19,  4, 16,  9, 21, 15,  3, 11, 20, 15,  0, 10, 25, 16
0, 49, 1, 0,  2, 48,  3, 49, 13, 51,  1,  1,  3, 50,  2,  0,  1, 60, 26
0, 50, 1, 0,  1, 51,  3, 50, 12, 51, 13,  1,  8, 49,  6,  0,  1, 62,  3
0, 55, 1, 0,  2, 54,  1, 55,  1, 53,  1,  1,  2, 54,  2,  0,  2, 54, 26

Antti Karttunen, Table of n, a(n) for n = 1..8256; the first 128 antidiagonals of the array

Cf. A269158 (transpose), A268729 (variant).

(define (A269159 n) (A269158auxbi (A004736 n) (+ -1 (* 2 (A002260 n))))) ;; Code for A269158auxbi given in A269158.

cp's OEIS Frontend

A268728 Square array A(row,col) = B(row,(2*col)-1), where B(p,2q-1) = 0 if gcd(p,2q-1) > 1, and A269158(p,q) otherwise. Array is read by descending antidiagonals as A(1,1), A(1,2), A(2,1), A(1,3), A(2,2), A(3,1), ...

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