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

Author

N. J. A. Sloane, Jun 19 2013

Keywords

• sign
• tabf

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.

Examples

			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;
  ...
		

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.

Crossrefs

Partial sums of rows of triangle in A226520.

See A226519 for another version.

Third and fourth columns give A226914, A226915.

See also A226523.

Cf. A165477 (131071st row), A165582.

Programs

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

Magma

A226518:= func< n,k | n eq 1 select k else  (&+[JacobiSymbol(j, NthPrime(n)): j in [0..k]]) >;
[A226518(n,k) : k in [0..NthPrime(n)-1], n in [1..15]]; // G. C. Greubel, Oct 05 2024

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 *)

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

SageMath

def A226518(n,k): return k if n==1 else sum(jacobi_symbol(j, nth_prime(n)) for j in range(k+1))
flatten([[A226518(n,k) for k in range(nth_prime(n))] for n in range(1,16)]) # G. C. Greubel, Oct 05 2024