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A063987 Irregular triangle in which n-th row gives quadratic residues modulo the n-th prime. 18
1, 1, 1, 4, 1, 2, 4, 1, 3, 4, 5, 9, 1, 3, 4, 9, 10, 12, 1, 2, 4, 8, 9, 13, 15, 16, 1, 4, 5, 6, 7, 9, 11, 16, 17, 1, 2, 3, 4, 6, 8, 9, 12, 13, 16, 18, 1, 4, 5, 6, 7, 9, 13, 16, 20, 22, 23, 24, 25, 28, 1, 2, 4, 5, 7, 8, 9, 10, 14, 16, 18, 19, 20, 25, 28, 1, 3, 4, 7, 9, 10, 11, 12, 16, 21, 25 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,4

COMMENTS

For n>=2, row lengths are (prime(n)-1)/2. For example, since 17 is the 7th prime number the length of row 7 is (17 - 1)/2 = 8. - Geoffrey Critzer, Apr 04 2015

LINKS

T. D. Noe, Rows n=1..100 of triangle, flattened

C. F. Gauss, Vierter Abschnitt. Von den Congruenzen zweiten Grades. Quadratische Reste und Nichtreste. Art. 97, in "Untersuchungen über die höhere Arithmetik", Hrsg. H. Maser, Verlag von Julius Springer, Berlin, 1889.

EXAMPLE

Mod the 5th prime, 11, the (11-1)/2 = 5 quadratic residues are 1,3,4,5,9 and the 5 non-residues are 2,6,7,8,10.

The irregular triangle T(n,k) begins  (Here P(n) is 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:    1

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

...  reformatted, - Wolfdieter Lang, Mar 06 2016

MAPLE

with(numtheory): for n from 1 to 20 do for j from 1 to ithprime(n)-1 do if legendre(j, ithprime(n)) = 1 then printf(`%d, `, j) fi; od: od:

MATHEMATICA

row[n_] := (p = Prime[n]; Select[ Range[p - 1], JacobiSymbol[#, p] == 1 &]); Flatten[ Table[ row[n], {n, 1, 12}]] (* Jean-François Alcover, Dec 21 2011 *)

PROG

(PARI) residue(n, m)=local(r); r=0; for(i=0, floor(m/2), if(i^2%m==n, r=1)); r

isA063987(n, m)=residue(n, prime(m)) /* Michael B. Porter, May 07 2010 */

(Python)

from sympy import jacobi_symbol as J, prime

def a(n):

    p=prime(n)

    return [1] if n==1 else list(filter(lambda i: J(i, p)==1, range(1, p)))

for n in xrange(1, 11): print a(n) # Indranil Ghosh, May 27 2017

CROSSREFS

Cf. A063988, A010379 (6th row), A010381 (7th row), A010385 (8th row), A010391 (9th row), A010392 (10th row), A278580 (row 23), A230077.

Sequence in context: A100353 A080508 A178141 * A236269 A010126 A021712

Adjacent sequences:  A063984 A063985 A063986 * A063988 A063989 A063990

KEYWORD

nonn,tabf,nice,easy

AUTHOR

Suggested by Gary W. Adamson, Sep 18 2001

EXTENSIONS

Edited by Wolfdieter Lang, Mar 06 2016

STATUS

approved

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Last modified December 13 01:33 EST 2017. Contains 295954 sequences.