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
(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
(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
CROSSREFS
KEYWORD
sign,tabf
AUTHOR
N. J. A. Sloane, Jun 19 2013
STATUS
approved