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A269595 Irregular triangle in which n-th row the gives quadratic residues prime(n)- m modulo prime(n), for m from {1, 2, ..., prime(n)-1}, in increasing order. 4
1, 2, 1, 4, 3, 5, 6, 2, 6, 7, 8, 10, 1, 3, 4, 9, 10, 12, 1, 2, 4, 8, 9, 13, 15, 16, 2, 3, 8, 10, 12, 13, 14, 15, 18, 5, 7, 10, 11, 14, 15, 17, 19, 20, 21, 22, 1, 4, 5, 6, 7, 9, 13, 16, 20, 22, 23, 24, 25, 28 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

The length of row 1 is 1 and of row n, n >= 2, is (prime(n)-1)/2, where prime(n) = A000040(n).

LINKS

Table of n, a(n) for n=1..60.

FORMULA

For n = 1, prime(1) = 2: 1, and for odd primes n >= 2: the increasing values of m from {1, 2, ..., p-1} with the Legendre symbol (-m/prime(n)) = + 1.

T(n, k) = prime(n) - A063987(n,(prime(n)-1)/2-k+1). k=1..(prime(n)-1)/2, for n >= 2, and T(1, 1) = 1.

EXAMPLE

The irregular triangle T(n, k) begins (P(n) is here prime(n)):

n, P(n)\k 1  2  3  4  5  6  7  8  9 10 11 12 13 14

1,   2:   1

2,   3:   2

3,   5:   1  4

4,   7:   1  2  4

5,  11:   1  3  4  5  9

6:  13:   1  3  4  9 10 12

7,  17:   1  2  4  8  9 13 15 16

8,  19:   1  4  5  6  7  9 11 16 17

9,  23:   1  2  3  4  6  8  9 12 13 16 18

10, 29:   1  4  5  6  7  9 13 16 20 22 23 24 25 28

...

MATHEMATICA

t = Table[Select[Range[Prime@ n - 1], JacobiSymbol[#, Prime@ n] == 1 &], {n, 10}]; Table[Prime@ n - t[[n, (Prime@ n - 1)/2 - k + 1]], {n, Length@ t}, {k, (Prime@ n - 1)/2}] /. {} -> 1 // Flatten (* Michael De Vlieger, Mar 31 2016, after Jean-Fran├žois Alcover at A063987 *)

CROSSREFS

Cf. A000040, A063987.

Sequence in context: A107640 A030065 A328676 * A055176 A118267 A324755

Adjacent sequences:  A269592 A269593 A269594 * A269596 A269597 A269598

KEYWORD

nonn,tabf,easy

AUTHOR

Wolfdieter Lang, Mar 06 2016

STATUS

approved

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Last modified January 24 10:01 EST 2022. Contains 350534 sequences. (Running on oeis4.)