A226518 Irregular triangle read by rows: T(n,k) = Sum_{i=0..k} Legendre(i,prime(n)).

0, 1, 0, 1, 0, 0, 1, 0, -1, 0, 0, 1, 2, 1, 2, 1, 0, 0, 1, 0, 1, 2, 3, 2, 1, 0, 1, 0, 0, 1, 0, 1, 2, 1, 0, -1, -2, -1, 0, -1, 0, 0, 1, 2, 1, 2, 1, 0, -1, 0, 1, 0, -1, -2, -1, -2, -1, 0, 0, 1, 0, -1, 0, 1, 2, 3, 2, 3, 2, 3, 2, 1, 0, -1, 0, 1, 0, 0, 1, 2, 3, 4, 3, 4, 3, 4, 5, 4, 3, 4, 5, 4, 3, 4, 3, 4, 3, 2, 1, 0

Offset

1,13

Comments

Strictly speaking, the symbol in the definition is the Legendre-Jacobi-Kronecker symbol, since the Legendre symbol is defined only for odd primes.

The classical Polya-Vinogradov theorem gives an upper bound.

There is a famous open problem concerning upper bounds on |T(n,k)| for small k.

References

R. Ayoub, An Introduction to the Analytic Theory of Numbers, Amer. Math. Soc., 1963; p. 320, Theorem 5.1.

Beck, József. Inevitable randomness in discrete mathematics. University Lecture Series, 49. American Mathematical Society, Providence, RI, 2009. xii+250 pp. ISBN: 978-0-8218-4756-5; MR2543141 (2010m:60026). See page 23.

Elliott, P. D. T. A. Probabilistic number theory. I. Mean-value theorems. Grundlehren der Mathematischen Wissenschaften, 239. Springer-Verlag, New York-Berlin, 1979. xxii+359+xxxiii pp. (2 plates). ISBN: 0-387-90437-9 MR0551361 (82h:10002a). See Vol. 1, p. 154.

Links

Alois P. Heinz, Rows n = 1..70, flattened

D. A. Burgess, The distribution of quadratic residues and non-residues, Mathematika 4, 1957, 106--112. MR0093504 (20 #28)

Wikipedia, Legendre symbol.

Example

Triangle begins:
[0, 1],
[0, 1, 0],
[0, 1, 0, -1, 0],
[0, 1, 2, 1, 2, 1, 0],
[0, 1, 0, 1, 2, 3, 2, 1, 0, 1, 0],
[0, 1, 0, 1, 2, 1, 0, -1, -2, -1, 0, -1, 0],
[0, 1, 2, 1, 2, 1, 0, -1, 0, 1, 0, -1, -2, -1, -2, -1, 0],
[0, 1, 0, -1, 0, 1, 2, 3, 2, 3, 2, 3, 2, 1, 0, -1, 0, 1, 0],
...

Maple

with(numtheory);
T:=(n, k)->add(legendre(i, ithprime(n)), i=0..k);
f:=n->[seq(T(n, k), k=0..ithprime(n)-1)];
[seq(f(n), n=1..15)];

Mathematica

Table[p = Prime[n]; Table[JacobiSymbol[k, p], {k, 0, p-1}] // Accumulate, {n, 1, 15}] // Flatten (* Jean-François Alcover, Mar 07 2014 *)

Prog

(PARI) print("# A226518 ");
cnt=1; for(j5=1, 9, summ=0; for(i5=0, prime(j5)-1, summ=summ+kronecker(i5, prime(j5)); print(cnt, "  ", summ); cnt++)); \\ Bill McEachen, Aug 02 2013
(Haskell)
a226518 n k = a226518_tabf !! (n-1) !! k
a226518_row n = a226518_tabf !! (n-1)
a226518_tabf = map (scanl1 (+)) a226520_tabf
-- Reinhard Zumkeller, Feb 02 2014

Crossrefs

Partial sums of rows of triangle in A226520.

Cf. A165582. See A226519 for another version.

Third and fourth columns give A226914, A226915. See also A226523.

Cf. A165477 (131071st row).

Sequence in context: A218880 A024356 A143947 * A337518 A073781 A048622

Adjacent sequences: A226515 A226516 A226517 * A226519 A226520 A226521

Keyword

sign,tabf

Author

N. J. A. Sloane, Jun 19 2013

Status

approved