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
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
Programs
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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
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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
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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)];
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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
Comments