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

1,5

Comments

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.

Links

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

Formula

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.

Example

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

Prog

(Scheme)
(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))))

Crossrefs

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.

Sequence in context: A358188 A117651 A373426 * A187196 A187197 A375368

Adjacent sequences: A268725 A268726 A268727 * A268729 A268730 A268731

Keyword

nonn,tabl

Author

Antti Karttunen, Feb 19 2016

Status

approved